Statistical mechanics of Beltrami flows in axisymmetric geometry: Theory reexamined
Aurore Naso, Romain Monchaux, Pierre-Henri Chavanis, Berengere Dubrulle
aa r X i v : . [ phy s i c s . f l u - dyn ] J u l Statistical mechanics of Beltrami flows in axisymmetric geometry:Theory reexamined
Aurore Naso ∗ , Romain Monchaux , Pierre-Henri Chavanis , and B´ereng`ere Dubrulle SPEC/IRAMIS/CEA Saclay, and CNRS (URA 2464),91191 Gif-sur-Yvette Cedex, France Laboratoire de Physique Th´eorique (UMR 5152),Universit´e Paul Sabatier, 118 route de Narbonne,31062 Toulouse, France (Dated: June 4, 2018)A simplified thermodynamic approach of the incompressible axisymmetric Euler equations is con-sidered based on the conservation of helicity, angular momentum and microscopic energy. Statisticalequilibrium states are obtained by maximizing the Boltzmann entropy under these sole constraints.We assume that these constraints are selected by the properties of forcing and dissipation. The fluc-tuations are found to be Gaussian while the mean flow is in a Beltrami state. Furthermore, we showthat the maximization of entropy at fixed helicity, angular momentum and microscopic energy isequivalent to the minimization of macroscopic energy at fixed helicity and angular momentum. Thisprovides a justification of this selective decay principle from statistical mechanics. These theoreticalpredictions are in good agreement with experiments of a von K´arm´an turbulent flow and provide away to measure the temperature of turbulence and check Fluctuation-Dissipation Relations (FDR).Relaxation equations are derived that could provide an effective description of the dynamics towardsthe Beltrami state and the progressive emergence of a Gaussian distribution. They can also providea numerical algorithm to determine maximum entropy states or minimum energy states.
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I. INTRODUCTION
In a turbulent flow, the number of degrees of freedomscales like Re / , where Re is the Reynolds number, andcan reach 10 for atmospheric-like flows, comparable tothe Avogadro number. This is beyond the present ca-pacity of computers. For example, numerical simulationsof a von K´arm´an (VK) turbulent flow at Re = 10 , astandard laboratory flow used for turbulence studies (seebelow), would require resolutions of the order of 10 gridpoints and integration times of the order 10 years of cpuwith current computers. This conclusion justifies the in-troduction of turbulence models to reduce the numberof degrees of freedom and make turbulence amenable tonumerical simulation or theoretical understanding. Thisgoal cannot be reached unless the different componentsof turbulence and their interactions are identified.Turbulence being intrinsically a stochastic process, itcan be decomposed in two components: the mean flowand the fluctuations around it. A good turbulence modelshould therefore be able to predict both the structure ofthe mean flow, and its influence on and through fluctu-ations, within a reduced number of degrees of freedom.This kind of information is typically provided by statisti-cal mechanics. Can we adapt statistical methods to dealwith the turbulence problem? ∗ Present address: Laboratoire de Physique, Ecole NormaleSup´erieure de Lyon and CNRS (UMR 5672), 46, all´ee d’Italie,69364 Lyon Cedex 07, France
This program has been pioneered by Onsager [1],Montgomery & Joyce [2] and Lundgren & Pointin [3] inthe framework of two-dimensional point vortices. In thelast decade, this statistical approach has been extendedby Miller [4] and Robert and Sommeria [5] to simplified2D or quasi 2D flows with continuous vorticity. Evenmore recently, Leprovost et al. [6] have shown that the2D formalism could actually be applied to a typical 2D1/2 situation, an axisymmetric flow. They obtained a re-lationship that gives the general shape of stationary solu-tions (mean flows) of the axisymmetric Euler equations.This relationship has been tested and confirmed experi-mentally in a turbulent von K´arm´an flow by Monchaux et al. [7] who observed that, at high Reynolds numbers,the selected shape is Beltrami, with vorticity and velocityaligned everywhere. As discussed in Appendix A 1, sucha shape cannot be obtained with the thermodynamicalapproach of Leprovost et al. [6]. In the present work, werevisit the theoretical tools in order to capture Beltramistates as statistical equilibrium states. We also extendthe computations one step further by considering fluctu-ations around mean field.Specifically, we develop a simplified thermodynamicapproach based on the conservation of helicity, angu-lar momentum and microscopic energy. We assume thatthese constraints are selected by the properties of forcingand dissipation. From a maximum entropy principle wederive the mean flow and the fluctuations around it. Wefind that the mean flow is in a Beltrami state and that thefluctuations are Gaussian. We also show that the maxi-mization of entropy at fixed helicity, angular momentumand microscopic energy is equivalent to the minimization of macroscopic energy at fixed helicity and angular mo-mentum.
This justifies from statistical mechanics a se-lective decay principle introduced previously from phe-nomenological arguments [8]. We use the mean field the-ory to link the fluctuations to the response of the meanflow to perturbations (susceptibility) and to the temper-ature in a way reminiscent of the Fluctuation-DissipationTheorem. This provides a way to measure the temper-ature of turbulence through the fluctuation level. Theanalogy with 2D turbulence is discussed. In fact, due tothe dual nature of axisymmetric flows, intermediate be-tween 2D and 3D turbulence, we find the emergence oftwo different effective temperatures in the fluctuations.One temperature, characterizing velocity fluctuations, isrelated to the formation of coherent structures like in 2Dturbulence. Another one, characterizing vorticity fluctu-ations, is related to 3D vorticity stretching and divergeswith increasing resolution. These predictions have beentested in companion papers [9, 10] based on PIV mea-surements in a turbulent von K´arm´an flow and are infair agreement with observations.The paper is organized as follows. In Section II, werecall the basic problematics associated with statisticalmechanics of turbulence, and formulate our hypothesesand the associated theoretical framework. In Section III,we recall the stationary solutions and the conservationlaws (the backbone of the statistical mechanics approach)of axisymmetric flows. In Sec. IV, we recall the phe-nomenological selective decay principle leading to Bel-trami flows. Section V is devoted to the computation ofthe statistical equilibrium states of axisymmetric inviscidflows using mean field theory. We derive the Gibbs statesand the fluctuation-dissipation relations (FDR) with twodifferent mean field approximations. In each case, themean flow is in a Beltrami state and the fluctuationsare Gaussian. In Section VI, we make the connectionbetween different variational principles that character-ize the equilibrium states. For each principle, we pro-pose a set of relaxation equations that can be used asa numerical algorithm to solve the variational problem.These relaxation equations can also provide an effectivedescription of the relaxation of the system towards maxi-mum entropy states. Finally, we justify through statisti-cal mechanics the phenomenological principle accordingto which: “the mean flow should minimize the macro-scopic energy at fixed helicity and angular momentum”.
II. HYPOTHESES AND THEORETICALFRAMEWORKA. Turbulence, Navier-Stokes equations andclassical statistical mechanics
A turbulent flow is described by the Navier-Stokesequations ∂ u ∂t + u · ∇ u = − ρ ∇ p + ν ∇ u + f, (1) where u is the velocity, p the pressure, ρ the fluid density, ν its kinematic viscosity, and f a forcing. In the absenceof forcing, the velocity decays to zero due to the dissipa-tion, so that turbulence is an intrinsic out-of-equilibriumproblem. In the sequel, we focus on the simplest situa-tion, where forcing and dissipation equilibrate on aver-age, so that stationary states can arise. The goal of thepresent paper is to describe these stationary states andthe fluctuations around them using tools borrowed fromclassical statistical mechanics. Specifically, we are goingto introduce a Hamiltonian system, perform equilibriumor near equilibrium statistical mechanics and compute itsequilibrium states. B. Stationary Navier-Stokes solutions vs solutionsof Euler equations
Since forcing and dissipation equilibrate on averagefor stationary solutions of the Navier-Stokes equations,it seems natural to consider this limiting case first in ourquest of a framework suitable for classical statistical me-chanics. In such a limit, we get the Euler equations ∂ u ∂t + u · ∇ u = − ρ ∇ p. (2)This is indeed a Hamiltonian system as long as one con-siders regular solutions such as those based on finiteGalerkin expansions. In 2D turbulence, the considera-tion of Euler solutions to describe Navier-Stokes station-ary solutions is well accepted, based on the remark thatthe vorticity cannot blow up and that the limit ν → ν → / stationary states and fluctuations of an out-of equi-librium system, the forced Navier-Stokes equations, canbe described by statistical equilibrium states and fluctua-tions of the Euler equations without forcing and dissipa-tion . C. The Euler system and conservation laws
The Euler equations for regular solutions are charac-terized by a number of conservation laws that depend onthe geometry and on the dimension of the system. In 2Dturbulence, for example, the conservation laws are thekinetic energy E = R u d x , the enstrophy Ω = R ω d x ,where ω z = ∇ × u is the vorticity, and, more generally,any function of the vorticity (Casimirs). In 3D turbu-lence, the generic conserved quantities are the kineticenergy E and the helicity H = R u · ω d x . Additionalconservation laws are possible in the presence of addi-tional symmetries, such as axisymmetry, see [6].In the presence of forcing and dissipation, these conser-vation laws are altered. In the sequel, we shall postulatethat the balance between forcing and dissipation selectssome particularly relevant conservation laws among theinfinity of inviscid invariants. In particular, we shall ar-gue that there exists relevant situations in which the onlyconserved quantities are the microscopic energy E andthe helicity H . This property holds, for example, for verysimple solutions of the Euler equations such that the ve-locity and the vorticity are aligned everywhere in the flow(Beltrami state). Our aim in this paper is not to deter-mine the mechanisms that select these invariants. Thisis a complicated problem that depends on the propertiesof forcing and dissipation and on the Reynolds number.However, to motivate our approach, we show in compan-ion papers [7, 9, 10] that our assumptions are consistentwith experimental results in the limit of large Reynoldsnumbers. D. Boundary conditions
When we try to model a turbulent flow by numericalsimulations, the choice of boundary conditions is a ma-jor and complicated problem. Indeed, the results turnout to depend sensitively upon what boundary conditionshave been chosen. For example, for two-dimensionalNavier-Stokes turbulence, qualitatively different resultshave been obtained for free-slip boundary conditions inrigid squares [12–14] and rectangles [2], for rectangu-lar periodic boundaries [8, 15, 16], for no-slip circularboundaries [17, 18], for free-slip circular boundaries [19],for stress-free circular boundaries [18], and for no-slipsquare boundaries [20]. For our problem, we argue thatnumerical simulations cannot deal with sufficiently highReynolds numbers (see the Introduction) and we ratherfocus on an experimental device, namely a von K´arm´anflow [7, 9]. In that case, the boundary conditions are automatically determined by the experimental geometryand the forcing.Since our theoretical approach is inviscid, we cannothope to describe the experimental flow close to its bound-aries. We rather claim that, when forcing and dissipationequilibrate each other, this flow can be reasonably de-scribed with an inviscid approach far enough from theboundaries. Our theory will be derived with the inviscidboundary conditions reflecting a cylindrical closed do-main ( i.e. ψ = 0 on the boundary). The comparisonwith experiments, detailed in [10], will be performed overa sub-domain of the experiment, far from boundary andforcing. As already shown in [7, 9], this restriction is suf-ficient to obtain good agreement between inviscid theoryand experimental fields, at the zeroth order approxima-tion ( i.e. if one considers the mean flow topology and itssmall fluctuations). III. EULER EQUATION IN THEAXISYMMETRIC CASEA. A convenient formulation of the axisymmetricEuler equations
In the axisymmetric case, the incompressible Eulerequations take the form1 r ∂∂r ( ru r ) + ∂u z ∂z = 0 , (3) ∂u r ∂t + u r ∂u r ∂r + u z ∂u r ∂z − u θ r = − ρ ∂p∂r , (4) ∂u θ ∂t + u r ∂u θ ∂r + u z ∂u θ ∂z + u r u θ r = 0 , (5) ∂u z ∂t + u r ∂u z ∂r + u z ∂u z ∂z = − ρ ∂p∂z , (6)where ( u r , u θ , u z ) are the velocity components in a cylin-drical referential ( r, θ, z ). Here, r runs from 0 to R and z from 0 to 2 h (we take the origin of the z axis at the bot-tom of the domain). Furthermore, we choose the lengthunit such that the total volume is unity: R rdrdz = 1(due to the axial symmetry, we systematically divide allthe volume integrals by 2 π ). It was shown in [6] thatthe axisymmetric incompressible Euler equations can berewritten in a simplified form in terms of σ , ξ and ψ ,where σ = ru θ is the angular momentum, ξ is the po-tential vorticity related to the azimuthal component ofthe vorticity by ξ = ω θ /r , and ψ is the streamfunctionassociated with the poloidal component of the velocity: u = u θ ˆe θ + ∇ × (cid:18) ψr ˆe θ (cid:19) . (7)Note that u r = − ∂ z ψ/r and u z = ∂ r ψ/r . The axisym-metric Euler equations can then be recast as [6]: ∂σ∂t + { ψ, σ } = 0 , (8) ∂ξ∂t + { ψ, ξ } = ∂∂z (cid:18) σ y (cid:19) , (9)∆ ∗ ψ ≡ y ∂ ψ∂z + ∂ ψ∂y = − ξ, (10)where y = r / { , } is the Poisson bracket ( { ψ, φ } = ∂ y ψ∂ z φ − ∂ z ψ∂ y φ ) and ∆ ∗ is a pseudo-Laplacian.A few general remarks are in order regarding this spe-cial case: i) One sees from Eq. (8) that the angular mo-mentum is conserved by the fluid particles and can onlybe mixed through the Euler dynamics. This is the ana-log of vorticity mixing in 2D turbulence and it justifiesthe introduction of a mixing entropy for the distributionof angular momentum (see Sec. V B). We will see thatwe can establish a close parallel with the statistical me-chanics of 2D turbulence to determine the distribution ofangular momentum. ii) By contrast, the potential vor-ticity is stirred like in 3D turbulence and not conserved.Therefore, the distribution of potential vorticity is moredifficult to investigate and this can lead to complicatedproblems such as cascade towards small scales, forma-tion of singularities etc. In fact, we shall show in thecompanion paper [21] that the statistical theory predictsthe existence of large-scale coherent structures (like in2D) but that these states are unstable saddle points andshould cascade towards smaller and smaller scales (likein 3D). However, these large-scale structures can have avery long lifetime because, being saddle points of entropy,they are unstable only for some particular perturbations.If the dynamics does not spontaneously generate theseoptimal perturbations, the system can remain “frozen” ina saddle point of entropy for a long time [21, 22]. There-fore, we are truly in a situation intermediate between 2Dturbulence and 3D turbulence. B. Stationary states
The axisymmetric Euler equations admit an infinitenumber of steady states. The general form of stationarysolutions of the axisymmetric Euler equations (8-10) hasbeen established in [6]. They are given by σ = f ( ψ ) , (11) − ∆ ∗ ψ = ξ = f ( ψ ) f ′ ( ψ )2 y + g ( ψ ) , (12)where f and g are arbitrary functions. When f is linear f ( x ) = λx and g = 0, the vorticity and velocity arealigned everywhere ω = λ u and the stationary flow is aBeltrami state. An alternative form also established in [6] is ψ = R ( σ ) , (13) ξR ′ ( σ ) − σ y = Q ( σ ) , (14)where R and Q are arbitrary functions. The relation be-tween Eqs. (11-12) and Eqs. (13-14) is developed in [6]provided that some invertibility properties for the func-tions are assumed. C. Conservation laws
Axisymmetric inviscid flows admit an infinite numberof conserved quantities, namely the total energy E = 12 Z ( u r + u θ + u z ) rdrdz = 12 Z ξψdydz + 14 Z σ y dydz, (15)the Casimirs I G = Z G ( σ ) dydz, (16)and the generalized helicities H F = Z ξF ( σ ) dydz, (17)where G and F are any (regular) functions. In the se-quel, we also introduce the notation H n and I n for thecase where F or G are power-laws x n . In particular, I = I = R σdydz (angular momentum), Γ = H = R ξdydz (circulation) and H = H = R ξσdydz (helicity) are con-served. D. Energy-helicity-Casimir functional
From the integral constraints discussed previously, ageneralization of the Arnol’d energy-Casimir functionalhas also been introduced in [6]. This is the energy-helicity-Casimir functional A = E + I G + H F . Considerthe optimization problemmin ξ,σ / max ξ,σ { A [ ξ, σ ] } . (18)A critical point of this functional determines a steadystate of the axisymmetric equations. Indeed, writing δA = δ ( E + I G + H F ) = 0 , (19)and taking variations on σ and ξ , we obtain ψ + F ( σ ) = 0 , (20) σ y + G ′ ( σ ) + ξF ′ ( σ ) = 0 , (21)and we recover the equations (13-14) characterizing asteady solution of the axisymmetric Euler equations. Thefact that we obtain all the steady states means that thequantities given by Eqs. (15,16,17) are the unique invari-ants of the axisymmetric incompressible Euler equations[6]. Furthermore, if the critical point of Eq. (18) is amaximum or a minimum of A then this steady state isnonlinearly dynamically stable. In many cases, we shallrestrict ourselves to formal nonlinear stability [23]. Weconsider small perturbations and we only require thatthe critical point is a (local) maximum or minimum of A such that the second order variations δ A = 12 Z δξδψ dydz + Z ( δσ ) y dydz + 12 Z G ′′ ( σ )( δσ ) dydz + 12 Z F ′′ ( σ ) ξ ( δσ ) dydz + Z F ′ ( σ ) δξδσ dydz, (22)are definite positive or definite negative for all perturba-tions δσ and δξ . Formal stability implies linear stability(in that case δ A can be used as a norm) but it does notimply nonlinear stability for infinite dimensional systems[23].On the other hand, the optimization problem given byEq. (18) provides just a sufficient condition of nonlineardynamical stability. More refined stability conditions canbe obtained by adding some constraints in the optimiza-tion problem [24, 25]. For example, the minimizationproblemmin ξ,σ { E [ ξ, σ ] | H F [ ξ, σ ] = H F , I G [ ξ, σ ] = I G } , (23)is more refined thanmin ξ,σ { E [ ξ, σ ] + µH F [ ξ, σ ] + αI G [ ξ, σ ] } , (24)in the sense that a solution of Eq. (24) is always a solu-tion of the more constrained problem given by Eq. (23),but the reciprocal may be wrong. This is similar to en-semble inequivalence in statistical mechanics where dif-ferent ensembles have the same critical points but notnecessarily the same maxima or minima (giving rise todifferent stability criteria) [25, 26]. Ensemble inequiva-lence is generic for systems with long-range interactionslike turbulence. E. Relaxation equations towards dynamicalequilibrium
We can introduce a set of relaxation equations thatsolve the optimization problem given by Eq. (18) byadapting the general methods described in [25, 27]. Wewrite the relaxation equations as ∂ξ∂t = X, ∂σ∂t = Y. (25) The time variations of A are given by˙ A = Z X ( ψ + F ( σ )) dydz + Z Y (cid:18) σ y + G ′ ( σ ) + ξF ′ ( σ ) (cid:19) dydz. (26)To determine the functions X and Y , we maximize therate of production (resp. dissipation) of A with the con-straints X ≤ C ξ , Y ≤ C σ . (27)This is the counterpart of Onsager’s linear thermody-namics. The variational principle can be written in theform δ ˙ A + Z χ δ (cid:18) X (cid:19) dydz + Z D δ (cid:18) Y (cid:19) dydz = 0 , (28)where χ and D are Lagrange multipliers associated withthe constraints given by Eqs. (27). This leads to therelaxation equations ∂ξ∂t = X = − χ [ ψ + F ( σ )] = − χ δAδξ , (29) ∂σ∂t = Y = − D (cid:20) σ y + G ′ ( σ ) + ξF ′ ( σ ) (cid:21) = − D δAδσ . (30)It is straightforward to establish that˙ A = − Z X χ dydz − Z Y D dydz. (31)Therefore, the relaxation equations (29,30) satisfy ˙ A ≤ D and χ are both positive and ˙ A ≥ D and χ areboth negative. On the other hand, ˙ A = 0 iff X = Y = 0so that ( ξ, σ ) is a steady state. By Lyapunov’s directmethod, we conclude that these equations can only con-verge towards a maximum of A (if D , χ are negative) or aminimum of A (if D , χ are positive). Saddle points of A are linearly unstable. Therefore, the relaxation equations(29,30) can be used as a numerical algorithm to solve theoptimization problem given by Eq. (18). IV. BELTRAMI FLOWS
Let us consider the minimization of energy at fixedhelicity and angular momentum [51]min ξ,σ { E [ ξ, σ ] | H, I } . (32)The critical points of this variational principle satisfy δE + µδH + αδI = 0 , (33)where µ and α are Lagrange multipliers. Taking the vari-ations of ξ and σ , we obtain ψ + µσ = 0 , (34) σ y + µξ + α = 0 . (35)These equations can be rearranged in the form σ = − µ ψ, (36) − ∆ ∗ ψ = ξ = ψ µ y − αµ . (37)They define a steady state of the axisymmetric Eulerequations of the form (11,12) with f ( x ) = − x/µ linearand g ( x ) = − α/µ constant. In that case, the vorticityand the (relative) velocity are aligned everywhere ω = − µ ( u + α e z × r ) , (38)and the stationary flow is a Beltrami state. This criticalpoint is a (local) minimum of energy at fixed helicity andangular momentum iff12 Z δξδψ dydz + Z ( δσ ) y dydz + µ Z δξδσ dydz ≥ , (39)for all perturbations δσ and δξ that conserve helicity andangular momentum at first order (see Appendix B).The variational problem given by Eq. (32) can be givenseveral justifications:(i) It can be introduced in a phenomenological man-ner from a selective decay principle [8]. Due to a smallviscosity, or other dissipative or relaxation mechanisms,the energy (fragile invariant) is dissipated while helicityand angular momentum (robust invariants) are approx-imately conserved. This selective decay principle has along history in physics. It first appeared in the MHDliterature with Taylor’s explanation of some behavior ofthe Zeta reversed-field pinch due to a conjectured rapiddecay of magnetic energy relative to magnetic helicity[28]. This principle leads to a force-free state, i.e. astate whose magnetic field is proportional to its owncurl [29, 30]. This is an analog of the Beltrami statesthat hydrodynamicists subsequently discovered in con-nection with axisymmetric turbulence. They are alsorelated to minimum enstrophy states in 2D turbulenceintroduced by Bretherton & Haidvogel [31] and later byLeith [32], leading to linear relationship between vorticityand stream function. Using the Chandrasekhar-Kendalleigenfunctions of the curl, these Beltrami states are easyto construct in both MHD and hydrodynamics. For ex-ample, they were used as a Galerkin basis for an exten-sive set of turbulent MHD computations by Shan et al. [33, 34].(ii) In Sec. VI, we shall propose a justification of theminimization problem given by Eq. (32) based on sta-tistical mechanics arguments. To our knowledge, this statistical mechanics justification has not been given be-fore.(iii) According to Eq. (23), the minimization problemgiven by Eq. (32) -if it has a solution- determines a steadystate of the axisymmetric Euler equations that is formallynonlinearly stable. Remark: the minimization problem given by Eq. (32)may not have a solution, i.e. a minimum of energyat fixed helicity and angular momentum may not exist.This is the conclusion that we shall reach in [21]. Theabsence of equilibrium state is usually associated witha “collapse” like the gravothermal catastrophe or theisothermal collapse in self-gravitating systems [35, 36]. Inthe present context, the “collapse” is associated with thebreak-up of large scale structures and the cascade of en-ergy at smaller and smaller scales. The relaxation equa-tions associated with the minimization problem given byEq. (32), derived in Sec. D 3, may give a qualitativeidea of how the system evolves by dissipating energy [21].However, since these equations are purely phenomenolog-ical, we stress that they may not necessarily provide anaccurate description of the true evolution of the system.
V. STATISTICAL MECHANICS OF THEAXISYMMETRIC EULER-BELTRAMI SYSTEMA. Basic set-up
In the previous sections, we considered steady states ofthe axisymmetric Euler equations. In a realistic situationwhere the system is forced and dissipated at small scales,these steady states describe the mean flow resulting fromthe balance between forcing and dissipation. However,there also exists fluctuations around the mean flow sothat the velocity field is u = u + u ′ , where u is the av-eraged velocity field and u ′ the fluctuations. We shallassume that u is axisymmetric and that the total sys-tem evolves while conserving the energy E = R u d r ,the helicity H = R u · ω d r and the angular momen-tum I = R ru θ d r , but no other constraint. We assumethat these conservation laws are selected by the proper-ties of forcing and dissipation, and consequently by theReynolds numbers. We shall call such flows an Euler-Beltrami system. It was found experimentally [7, 9]that the system approaches a Beltrami state when theReynolds number is sufficiently large, giving support tothe basic assumption of our theory.In the sequel, it will prove useful to operatea poloidal/toroidal decomposition such that u p =( u r , , u z ) and u t = (0 , u θ , u = u p + u t while the helicitydensity u · ω = u p · ω p + u t · ω t . For axisymmetric fields, R u p · ω p d r = R u t · ω t d r . To determine the distributionof angular momentum σ and vorticity ξ , we shall use aMean Field Theory (MFT). This method is traditionallyvery efficient in systems of high dimensionality, or withlong-range interactions, a condition met in fluid mechan-ics. In our system, we have at our disposal two privilegeddirections: the toroidal direction and the poloidal direc-tion. We therefore derive two different MFT procedures,freezing the fluctuations in one of the two directions tocapture the fluctuations in the other direction. In eachcase, we introduce a suitable entropy, and maximize itunder the energy, helicity and angular momentum con-straints so as to obtain the Gibbs states. From theseGibbs states, we derive relations for the mean flow andfor the fluctuations. The first approach, which is closelyrelated to the approach in 2D turbulence will give usthe mean field (Beltrami) and the distribution of angu-lar momentum (Gaussian). It will allow us to justify aprinciple of minimum energy at fixed helicity and angularmomentum. The second approach will give us the samemean field and the distribution of vorticity (Gaussian).Fluctuations are however found to diverge in the limit ofnumber of modes going to infinity, a pathology that canbe traced back to vorticity stretching. B. Mean field approximation on the poloidal field:the distribution of angular momentum σ
1. Computations
Let us first assume that the fluctuations are mainly inthe toroidal direction, so that the poloidal fluctuationscan be ignored | u ′ p | ≪ | u p | . In that case, the poloidalfield is only determined by ξ = ξ and ψ , so that u ′ ismade only by fluctuations of σ . Let us introduce thedensity probability ρ ( r , η ) to measure σ = η at position r = ( y, z ). Then, the local moments of the angular mo-mentum are σ n = R ρη n dη . To proceed further, we needto introduce an entropy. Since the angular momentumdensity σ is conserved by the flow but undergoes a com-plicated mixing process (like the vorticity in 2D), it isnatural to introduce the mixing entropy S [ ρ ] = − Z ρ ln ρ dydzdη, (40)similar to the one introduced by Miller-Robert-Sommeriain 2D turbulence. We expect the entropy to increaseduring the dynamics (while the helicity, the angular mo-mentum and the microscopic energy are conserved) untilthe flow achieves a steady state. The functional given byEq. (40) can also be interpreted as the neg-information(the opposite of the information). Maximizing this neg-information under given constraints is the simplest proce-dure we can adopt to compute the fluctuations, accordingto the information theory and its application to statisti-cal mechanics developed by Jaynes [37]. In our approach, the conserved quantities are E f.g. = 12 Z ξ ψ dydz + Z σ y dydz = 12 Z ξ ψ dydz + Z ρ η y dydzdη, (41) H = Z ξ σ dydz = Z ξρη dydzdη, (42) I = Z σ dydz = Z ρη dydzdη. (43)The first constraint given by Eq. (41) will be called themicroscopic (or fine-grained) energy because it takes intoaccount the fluctuations of σ . It is different from themacroscopic (or coarse-grained) energy E c.g. = 12 Z ξ ψ dydz + Z σ y dydz, (44)which ignores these fluctuations. We have E f.g. = E c.g. + E fluct . In our terminology, the energy will becalled a fragile constraint because it cannot be expressedin terms of the coarse-grained field since σ = σ . Whilethe microscopic energy E f.g. is conserved, the macro-scopic energy E c.g. is not conserved and can decay. Thereis the same distinction between the fine-grained enstro-phy Γ f.g. = R ω d r and the coarse-grained enstrophyΓ c.g. = R ω d r in 2D turbulence [22, 38]. On the otherhand, the helicity given by Eq. (42) and the angularmomentum given by Eq. (43) will be called robust con-straints because they can be expressed in terms of thecoarse-grained fields. We shall come back to this impor-tant distinction in Sec. VI.The most probable distribution at metaequilibrium isobtained by maximizing the mixing entropy S [ ρ ] at fixed E f.g. , H , I and local normalization R ρdη = 1. Introduc-ing Lagrange multipliers, the variational principle can bewritten as δS − β ξ δE f.g. − µ ξ δH − α ξ δI − Z ζ ( r ) δ (cid:18)Z ρdη (cid:19) d r = 0 , (45)where β ξ is the inverse temperature and µ ξ the helicalpotential (we have written these quantities with a sub-script ξ to recall that the fluctuations of ξ are ignored inthe present approach). The variations on ξ imply β ξ ψ + µ ξ σ = 0 , (46)while the variations on ρ yield the Gibbs state ρ ( r , η ) = 1 Z e − βξη y − ( µ ξ ξ + α ξ ) η , (47)where the “partition function” is determined via the nor-malization condition Z ( r ) = Z e − βξη y − ( µ ξ ξ + α ξ ) η dη. (48)From Eq. (47), the local average of the angular momen-tum is σ = − yβ ξ ( µ ξ ξ + α ξ ) . (49)Together with Eq. (46), this equation determines a Bel-trami state. On the other hand ρ ( r , η ), the distribution ofthe fluctuations of σ , is Gaussian with centered variance σ ≡ σ − σ = 2 yβ ξ . (50)The Gibbs state can be rewritten ρ ( r , η ) = (cid:18) β ξ πy (cid:19) / e − βξ y ( η − σ ) . (51)Therefore, our statistical theory based on the conserva-tion of E f.g. , H and I predicts that the mean flow isa Beltrami state with Gaussian fluctuations of angularmomentum.Note that Eq. (50) means that the toroidal velocityfluctuations are uniform u θ − u θ = 1 β ξ . (52)Therefore, β ξ can be interpreted as an inverse tempera-ture measuring the fluctuations of u θ . These predictionsenable the measurements of effective temperatures of tur-bulence through fluctuations of u θ in a Beltrami flow. Be-cause variances are positive, β ξ is always positive, unlikein the 2D situation where the temperature can be neg-ative (in the present context, the inverse temperature isthe equivalent of the Lagrange multiplier associated withthe conservation of microscopic enstrophy in 2D turbu-lence, which is positive [22]) . Note that Eq. (52) predicts uniformity of azimuthal velocity fluctuations which is aninteresting prediction of our theory. This has been con-firmed experimentally in [9].On the other hand, the energy contained in the fluctu-ations is simply E fluct = Z ρ ( η − σ ) y dydzdη = Z σ y dydz = 12 β ξ . (53)Therefore, the statistical temperature β − ξ can also beinterpreted as the energy (by unit volume) of the toroidalfluctuations. Moreover, for simple Beltrami flows with α ξ = 0, there is equipartition between the macroscopicenergy in the poloidal and toroidal directions E pc.g. ≡ Z ξ ψ dydz = Z σ y dydz ≡ E tc.g. , (54)with a simple connection with the helicity as E pc.g. = − µ ξ β ξ H. (55)
2. Comments
The statistical equilibrium state given by Eqs. (46) and(49) is of the form of Eqs. (11) and (12) where f is linear f ( x ) = λx (with λ = − β ξ /µ ξ ) and g is constant (with g = − α ξ /µ ξ ). This means that the equilibrium state isa stationary solution of the axisymmetric Euler equationand takes the shape of a Beltrami state (see Eq. (38)).We can also provide an interesting interpretation ofour fluctuation relation Eq. (52), predicting uniformityof azimuthal velocity fluctuations. This equation showsthat the azimuthal velocity fluctuations define an effec-tive statistical temperature 1 /β ξ . This equation may beregarded as formally analogous to a Fluctuation Dissipa-tion Relation (FDR) since it links fluctuations and tem-perature. These predictions enable the measurements ofturbulence effective temperatures through fluctuations of u θ in a Beltrami flow. As discussed previously, β ξ is al-ways positive. In contrast, µ ξ can take positive or nega-tive values, depending on the helicity sign.The analogy between our predictions and FDRs can ac-tually be pushed forward. Indeed, another possible wayto derive Eq. (52) is to introduce, as in classical statis-tical mechanics, the partition function Z describing theBeltrami equilibrium state in the mean field approxima-tion: σ − σ = 1 µ ξ δ log Zδξ = − µ ξ δσδξ , (56)where δ stands for functional derivative. Formally, themathematical object δσ/δξ can be seen as a responsefunction. With this point of view, Eq. (56) again reflectsa formal analogy with FDRs since another classical wayto write it down is to link the fluctuations of a field toits response to a perturbation. C. Mean field approximation in the toroidaldirection: the distribution of vorticity ξ
1. Spectral approach
We now assume that the fluctuations in the toroidaldirection are frozen so that σ = σ , and that they do notdepend on the azimuthal direction. Since the vorticity ξ is not conserved, we cannot in principle rigorously applya statistical mechanics to the fluctuations of vorticity.We present here a phenomenological approach, based onneg-information rather than mixing entropy, and will testits relevance by comparison with experimental data incompanion papers [9, 10].In our approach, the conserved quantities are E = 12 Z ψξdydz + Z σ y dydz, (57) H = Z σ ξdydz. (58) I = Z σ dydz. (59)In the expression of the energy, we note that the fluctua-tions of angular momentum have been neglected so that σ = σ while the fluctuations of potential vorticity havebeen taken into account so that ψξ = ψ ξ . In order to dealwith the term ψξ that introduces a nonlocality, we shalldevelop the statistical theory in the spectral space by us-ing an approach similar to that developed by Kraichnan[39] and Salmon et al. [40] in 2D turbulence. Let usfirst note that the field φ = ψ/r satisfies the differentialequation L φ ≡ − ∆ φ + 1 r φ = rξ = ω θ . (60)To solve the problem, we decompose the fields onto theeigenfunctions φ mn of L defined by L φ mn ≡ − ∆ φ mn + φ mn r = B mn φ mn , (61)with φ mn = 0 on the boundary. Taking the origin of the z axis at the bottom of the domain, the eigenfunctionsare given by the Hankel-Fourier modes φ mn = s hR J ( j m ) J (cid:18) j m rR (cid:19) sin (cid:16) nπz h (cid:17) , (62)where j m is the m th zero of Bessel function J . Themode φ mn corresponds to m cells in the radial directionand n cells in the vertical direction. The correspondingeigenvalues are B mn = (cid:18) j m R (cid:19) + (cid:16) nπ h (cid:17) . (63)The eigenfunctions are orthogonal with respect to thescalar product h f g i ≡ Z R Z h r drdz f g, (64)so that h φ mn φ m ′ n ′ i = δ mm ′ δ nn ′ where δ ij is the Kro-necker symbol. We now decompose the fields on theseeigenmodes writing ψr = φ = N m X m =1 N n X n =0 Ψ mn φ mn , (65) ω θ = rξ = N m X m =1 N n X n =0 ω mn φ mn , (66) u θ = σr = N m X m =1 N n X n =0 u mn φ mn , (67) where we have restricted the sum over finite number ofmodes, so as to respect Hamiltonian condition for theEuler equation. Moreover, the finite number of modesimplies a coarse graining of the solution, that is desirableto reach a stationary state. With this decomposition,using Eq. (60), we have by construction ω mn = B mn Ψ mn . (68)Inserting the decomposition given by Eqs. (65-67) in theexpression of the energy and helicity, and using the or-thogonality condition given by Eq. (64) and the Parsevalidentities, we obtain E = 12 X mn ω mn B mn + u mn ! , (69) H = X mn ω mn u mn , (70) I = X mn u mn h rφ mn i , (71)where the brackets denote a domain average. To applythe statistical theory, we introduce the density probabil-ity ρ mn ( ν ) of measuring the value ω mn = ν of the vortic-ity in the mode ( m, n ). We can rewrite the constraintsin the form E = 12 X mn Z (cid:18) ν B mn + u mn (cid:19) ρ mn ( ν ) dν, (72) H = X mn Z ω mn u mn ρ mn ( ν ) dν, (73) I = X mn u mn h rφ mn i . (74)
2. The multi-modes case
We first consider the situation where the energy isspread all over the different wavenumbers ( n, p ). In thatcase, there is no reason to specialize a special mode andwe can use the neg-information in spectral space to definethe entropy S = − X mn Z ρ mn ( ν ) ln ρ mn ( ν ) dν. (75)The Gibbs state is obtained by maximizing S at fixedenergy, helicity, angular momentum and normalization R ρ mn dν = 1. We write the variational principle as δS − β σ δE − µ σ δH − α σ δI − X mn χ mn δ (cid:18)Z ρ mn dν (cid:19) = 0 , (76)where β σ , µ σ and α σ are Lagrange multipliers (the sub-script σ recalls that the fluctuations of angular momen-tum have been neglected). The variations on u mn yield β σ u mn + µ σ ω mn + α σ h rφ mn i = 0 . (77)0Multiplying by φ mn , summing on the modes and using r = P mn h rφ mn i φ mn , we obtain β σ σ y + µ σ ξ + α σ = 0 . (78)The variations on ρ mn yield the Gibbs state ρ mn ( ν ) = 1 Z mn e − (cid:18) βσν B mn + µ σ u mn ν (cid:19) , (79)where Z mn is a factor ensuring the local normalizationcondition. The distribution of the fluctuations of ω mn istherefore Gaussian with β σ ω mn B mn + µ σ u mn = 0 , (80) ω mn − ω mn = B mn β σ . (81)According to Eq. (81), the mean fluctuating energy permode number ( ω mn − ω mn ) /B mn is constant: we haveequipartition of energy for the poloidal fluctuations. Thisresult is a classical outcome of equilibrium statistical me-chanics. Multiplying Eq. (80) by φ mn , summing on themodes and using Eq. (68), we obtain β σ ψ + µ σ σ = 0 . (82)Equations (78) and (82) show that the mean flow associ-ated with the statistical equilibrium state is a stationarysolution of the axisymmetric Euler equations correspond-ing to a Beltrami state. One can also deduce from Eq.(81) that the total volumic energy contained in the fluc-tuations is E fluct = X mn Z ( ν − ω mn ) B mn ρ mn ( ν ) dν = N tot β σ , (83)where N tot is the total number of modes. The lineardivergence with N tot comes from energy equipartition.Therefore, the statistical temperature in this case is pro-portional to the poloidal energy of fluctuations. This istherefore analog to the previous mean field case.It is also interesting to compute the azimuthal vorticityfluctuations. They are given by ω θ − ω θ = 1 β σ X mn B mn φ mn , (84)since ω mn and ω m ′ n ′ are independent if ( m, n ) = ( m ′ , n ′ ).For comparison with real turbulent data fields, we haveconstructed synthetic instantaneous fields obeying Eqs.(78), (79) and (82) to study some of their properties. Anexample is shown in Fig. 1. The average velocity field isobtained as follows: we choose a given number of modesto represent the field ( N m , N n ). Then, we get the coeffi-cient of the Hankel-decomposition u mn by a least-squarefit to u θ of an actual mean turbulent field (here, the ve-locity field obtained by counter-rotation at F = 6 Hz of TM73 impellers-this field is described in [9]). This field isshown in Fig. 1. Once the u mn are obtained, we get thecoefficient Ψ mn of the Hankel-decomposition of the meanstream function φ from a least-square fit to the poloidalexperimental velocity fields u r and u z using Eqs. (7).Such a field, shown in Fig. 1, obeys the relation (82)with B = − β σ /µ σ = − .
6, as shown in Fig. 2. Wethen obtain the mean azimuthal vorticity ω θ thanks to(60). It is shown in Fig. 1. We also use the modes ofthe decomposition to compute the theoretical variance ω rms following Eq. (84), shown in Fig. 2. Finally, wecompute an instantaneous fluctuation field of azimuthalvorticity by drawing for each mode a realization of theGaussian distribution Eq. (79) and reconstructing thefield through Eq. (66). The results are provided in Fig.1. We see that the theoretical variance ω rms is not inde-pendent of r and z but its dependence in z is weak. Inthe radial direction, it oscillates mildly around a valueΩ ∞ : ω θ − ω θ ≈ Ω ∞ . (85)Noteworthy, Ω ∞ is much larger than the value B /β σ obtained in the one mode case (see Eq. (100) of Sec.V C 3). This is reminiscent of what has been observed inreal turbulent data fields [9]. It is therefore interestingto study further the dependence of Ω ∞ with respect tothe number of modes of the problem. We have foundempirically that this value behaves likeΩ ∞ = N tot h B i β σ , (86)where h B i is a mean Beltrami factor, defined as: h B i ≡ N tot X mn B mn . (87)This number depends on the set of modes (values of n and m ) considered. For example, in an isotropic situationwhen one sums only over modes such that n = m , B mn ∼ n and h B i ∼ N − tot N n ∼ N / tot (here N tot = N n N m = N n ). In an anisotropic situation such as N m = 1, B mn ∼ n and h B i ∼ N − tot N n ∼ N tot (here N tot = N n N m = N n ).Integrating Eq. (84) over the volume, we get the az-imuthal enstrophy fluctuationsΩ fluct = Z ( ω θ − ω θ ) dydz = 1 β σ X mn B mn = N tot β σ h B i . (88)Since h B i grows algebraically with N tot (see above), theenstrophy fluctuations therefore diverge as N tot × N αtot when the number of modes becomes infinite. The lin-ear part of the divergence is comparable with the lineardivergence obtained for the variance of energy fluctua-tions, and can be thought to be an outcome of equilib-rium statistical mechanics. Note that in real turbulent1 FIG. 1: (Color online) Example of synthetic instantaneousfields in the multi-mode case, at − β σ /µ σ = − . β σ = 2:(a) projection of ( u r , u θ , u z ) over a vertical meridional plane.The toroidal velocity is coded in color, the poloidal velocityis denoted with arrows; (b) projection of φ over a verticalmeridional plane; (c) projection of ω θ over a vertical merid-ional plane; (d) projection of ω ′ θ over a vertical meridionalplane. The field is constructed with N m = 2 radial modesand N m = 2 axial modes. The amplitude of the velocitymodes results from a least square fit to the average velocityfield described in [9], with a forcing using TM73 turbines incontra-rotation at F = 6 Hz over eigenmodes given by Eqs.(61). The instantaneous value is obtained by drawing onerealization of random vorticity fluctuations according to thetheoretical Gibbs distribution given by Eq. (79). flows, -out-of-equilibrium solutions with non-zero energyflux-, the energy of fluctuations by mode E fluct /N tot re-mains finite but the enstrophy of fluctuations by modeΩ fluct /N tot ∼ N αtot diverges algebraically with the num-ber of modes in 3D, while the divergence is much milder(logarithmic) in 2D. This difference can be seen as thesignature of 3D vortex stretching, that is captured byour model. Indeed, vortex stretching induces a trans-fer of vorticity at continuously decreasing scales, therebyleading to an enstrophy divergence.Note that both the enstrophy fluctuations and the en-ergy fluctuations provide an estimate of the statisticaltemperature. The comparison of temperature in betweenthe two measurements actually provides a measure of thenumber of modes in the system since energy fluctuationsare proportional to N tot and enstrophy fluctuations be-have roughly like N . tot for an isotropic situation. Thiswill be further discussed in Sec. V C 4.
3. The one mode case
The previous Gibbs distributions do not couple modeswith different wavenumbers. They are the analogs ofthe Boltzmann laws found in simple quasi-geostrophicmodels [40] in the case where energy is spread evenlyover all modes (an implicit assumption behind our choice −1 −0.5 0 0.5 1−4−2024 Φ u θ (a) ω r m s (b) FIG. 2: (a) (Color online) Toroidal velocity u θ as a functionof the stream function φ . The line is a linear fit, with slope B = − .
6. (b) Spatial variation of the theoretical vorticityvariance as a function of r only. This variance has been ob-tained using the modes of the Hankel decomposition of thevelocity field and Eq. (84). The dispersion in vertical di-rection corresponds to a dispersion of variance at a given r with varying z . The horizontal line is the empirical value N tot h B i /β σ . The conditions are the same as in Fig. 1. of entropy). It is interesting to consider the oppositecase, where all the energy is concentrated in one mode( m c , n c ). The results pertaining to this case have beendiscussed in [9]. We present here the corresponding de-tailed computation. In such a case, it is easy to checkthat ψ = 2 yξB + κy, (89)where we have noted B ≡ B m c n c for brevity. Fur-thermore, the term κy can always be introduced sinceit is in the kernel of ∆. Since the energy is concen-trated in one mode in the spectral space, we cannotuse the neg-information anymore in that space to de-termine the Gibbs distribution. However, by the Lin-quist theorem, such a peaked probability in the spectralspace corresponds to a spread probability in the physicalspace. We therefore turn back to the density probabil-ity ρ ( r , ν ) to measure ξ = ν at position r , introduce theneg-information S [ ρ ] = − Z ρ ln ρ dydzdν, (90)in the physical space, and consider the maximization of S [ ρ ] at fixed E , H and normalization R ρ dν = 1. UsingEq. (89), the constraints can be written E = Z yB ξ dydz + κ Z ξy dydz + Z σ y dydz = Z yB ρν dydzdν + κ Z yρν dydzdν + Z σ y dydz, (91) H = Z σ ξ dydz = Z σρν dydzdν, (92) I = Z σ dydz. (93)2We write the variational principle as δS − β σ δE − µ σ δH − α σ δI − Z χ ( r ) δ (cid:18)Z ρ dν (cid:19) d r = 0 . (94)The variations on σ imply β σ σ y + µ σ ξ + α σ = 0 , (95)and the variations on ρ yield the Gibbs state ρ ( r , ν ) = 1 Z e − βσyB ν − ( µ σ σ + β σ κy ) ν , (96)where Z is the normalization factor. The distribution istherefore Gaussian. Its first two moments are ξ = − B µ σ β σ σ y − B κ , (97) ξ ≡ ξ − ξ = B yβ σ . (98)The Gibbs state can be rewritten ρ ( r , ν ) = (cid:18) yβ σ πB (cid:19) / e − yβσB ( ν − ξ ) . (99)One sees that relations (97) and (95) can be satisfied si-multaneously only if B = ( β σ /µ σ ) and κ = 4 α σ µ σ /β σ .This fixes the wavenumbers ( m c , n c ) of the mode in whichenergy is accumulated and provides a physical interpreta-tion of the ratio β σ /µ σ . The relations (97) and (89) showthat the mean flow associated with the statistical equi-librium state is a stationary solution of the axisymmetricEuler equation corresponding to a Beltrami state. More-over, Eq. (98) shows that the fluctuations of vorticity areuniform and scale as ω θ − ω θ = B β σ . (100)Integrating over the volume, we find that the azimuthalenstrophy fluctuations areΩ fluct = Z ( ω θ − ω θ ) dydz = 1 β σ B . (101)We may also compute the energy contained in the fluc-tuations as E fluct = Z y ( ν − ξ ) B ρdydzdη = Z yB ξ dydz = 12 β σ . (102)Therefore, the statistical temperature can be sim-ply interpreted here as the volumic energy of thepoloidal fluctuations. Note that in such a simple case,Ω fluct /E fluct = B , consistent with the multi-mode re-sult since B = h B i in the “one mode case”. Moreover, for simple Beltrami flows with α σ = 0, there is equiparti-tion between the energy of the mean flow in the poloidal E p and toroidal direction E t : E c.g.p ≡ Z ξ ψ dydz = Z σ y dydz ≡ E c.g.t , (103)with a simple connection with the mean helicity as E c.g.p = − µ σ β σ H. (104)This case is therefore the analog to the mean field toroidalcase, with uniformity of fluctuations and simple connec-tion with helicity.
4. A note on vorticity fluctuations
The link of the present results with experiments hasbeen partially discussed in [7, 9], with focus on the meanfield toroidal case and the one-mode mean field poloidalcase. A detailed comparison is provided in a compan-ion paper [10]. It is however interesting to come back toone puzzling result of [9] to give it a new interpretationin the present context. It has indeed been found thatin a turbulent counter-rotating von K´arm´an flow boththe velocity and vorticity fluctuations are approximatelyuniform over the box. When interpreted in the contextof the mean field toroidal case, and the one-mode meanfield poloidal case, the value of this constant providesan estimate of the two inverse temperatures β ξ and β σ through Eqs. (52) and (100). Experimentally, one finds β ξ ≪ β σ , with a ratio β ξ /β σ ranging from 8 to 17 indifferent forcing configurations. In the restricted contextdiscussed in [9], this difference is puzzling, and pointstowards the existence of two different temperatures. Ifhowever one considers a wider context, in which the vor-ticity fluctuations are considered to span several modes,the experimental measurements allow for another inter-pretation. Indeed, using our least-square fitting to exper-imental data with varying N n and N m with N n = N m (isotropic case), we found that ω θ − ω θ ≈ N . tot . β σ , in-stead of the value 3 . /β σ predicted in the one-mode case N tot = 1 (recall that B = − . β ξ = β σ instead of N tot = 1, we canthen use the experimental measurements to infer N tot .One finds a value ranging from N tot = 4 to N tot = 6.Such a small number is intriguing because the usual be-lief is that turbulent flows are characterized by a verylarge number of degrees of freedom. There are howeverother indications ( e.g. in the dynamo context [41]) thatturbulent flows can be described using tools adapted fromdynamical systems, as if the effective number of degreesof freedom were indeed small.3 VI. CONNECTION BETWEEN DIFFERENTVARIATIONAL PRINCIPLES
In this section, we make the connection between dif-ferent variational principles that characterize the equilib-rium states. For each principle, we propose a relaxationequation (see Appendix D) that can be used as a nu-merical algorithm to solve the corresponding variationalproblem. These relaxation equations can also providean effective description of the relaxation of the systemtowards the equilibrium state. They will be solved nu-merically in [10]. We finally justify through statisticalmechanics the phenomenological principle according towhich: “the mean flow should minimize the macroscopicenergy at fixed helicity and angular momentum”. In thissection, we ignore the fluctuations of vorticity and exclu-sively consider the mean field theory of the poloidal fielddeveloped in Sec. V B.
A. The basic variational principle
The basic maximization problem that we have to solveis: max ρ,ξ { S [ ρ ] | E f.g. , H, I, Z ρdη = 1 } , (105)with S [ ρ ] = − Z ρ ln ρ dydzdη, (106)and E f.g. = 12 Z ξ ψ dydz + Z σ y dydz = 12 Z ξ ψ dydz + Z ρ η y dydzdη, (107) H = Z ξ σ dydz = Z ξρη dydzdη, (108) I = Z σ dydz = Z ρη dydzdη. (109)The critical points are determined by the variationalprinciple δS − βδE f.g. − µδH − αδI − Z ζ ( r ) δ (cid:18)Z ρdη (cid:19) dydz = 0 . (110)The variations on ξ imply βψ + µσ = 0 , (111)while the variations on ρ yield the Gibbs state ρ ( r , η ) = (cid:18) β πy (cid:19) / e − β y ( η − σ ) , (112) with σ = − yβ ( µξ + α ) , (113) σ ≡ σ − σ = 2 yβ . (114)As already indicated, the last relation shows that thetemperature must be positive. Furthermore, a criticalpoint of (105) is an entropy maximum at fixed E f.g. , H , I and normalization iff δ J ≡ − Z ( δρ ) ρ dydzdη − β Z δξδψ dydz − µ Z δξδσ dydz ≤ , (115)for all perturbations δρ and δξ that conserve energy, he-licity, angular momentum and normalization at first or-der . B. An equivalent but simpler variational principle
The maximization problem (105) is difficult to solvebecause the stability condition (115) is expressed in termsof the distribution ρ ( r , η ). We shall here introduce anequivalent but simpler maximization problem by “pro-jecting” the distribution on a smaller subspace. To solvethe maximization problem (105), we can proceed in twosteps [25].(i) First step: we first maximize S at fixed E f.g. , H , I , R ρ dη = 1 and σ ( r ) = R ρη dη and ξ ( r ). Since the spec-ification of σ ( r ) and ξ ( r ) determines R ψξ d r , H and I ,this is equivalent to maximizing S at fixed R ρ η y dydzdη , R ρ dη = 1 and σ ( r ) = R ρη dη . Writing the variationalproblem as δS − βδ (cid:18)Z ρ η y dydzdη (cid:19) − Z λ ( r ) δ (cid:18)Z ρη dη (cid:19) dydz − Z ζ ( r ) δ (cid:18)Z ρ dη (cid:19) dydz = 0 , (116)we obtain ρ ( r , η ) = (cid:18) β πy (cid:19) / e − β y ( η − σ ) , (117)and we check that it is a global entropy maximum withthe previous constraints since δ S = − R ( δρ ) ρ dydzdη ≤ ρ so their second variationsvanish). We also note that the centered local variance ofthis distribution ρ is σ ≡ σ − σ = 2 yβ , (118)implying β ≥
0. Using the optimal distribution given byEq. (117), we can now express the functional in terms4of ξ , σ and β writing S = S [ ρ ] and E f.g. = E f.g. [ ρ ].After straightforward calculations, we obtain S = −
12 ln β, (119) E f.g. = 12 Z ξψ dydz + 12 β + Z σ y dydz, (120) H = Z ξ σ dydz, (121) I = Z σ dydz, (122)up to some constant terms. Note that β is determinedby the energy constraint given by Eq. (120) leading to12 β = E f.g. − E c.g. [ ξ, σ ] , (123)where E c.g. is the macroscopic energy defined by Eq.(44). This relation can be used to express the entropyEq. (119) in terms of ξ and σ alone.(ii) Second step: we now have to solve the maximiza-tion problem max σ,ξ { S [ σ, ξ ] | E f.g. , H, I } , (124)with S = 12 ln (cid:18) E f.g. − Z ξψ dydz − Z σ y dydz (cid:19) , (125) H = Z ξ σ dydz, (126) I = Z σ dydz. (127)(iii) Conclusion:
Finally, the solution of (105) is givenby Eq. (117) where σ is solution of (124). Therefore,(105) and (124) are equivalent but (124) is easier to solvebecause it is expressed in terms of σ and ξ while (105) isexpressed in terms of ρ and ξ .Up to second order, the variations of entropy given byEq. (125) are∆ S = − β (cid:18) Z ψδξ dydz + Z δψδξ dydz + Z ( δσ ) y dydz + Z σδσy dydz (cid:19) − β (cid:18)Z ψδξ dydz + Z σδσ y dydz (cid:19) , (128) where β is given by Eq. (123). The critical points of(124) are determined by the variational problem δS − µδH − αδI = 0 . (129)The variations on ξ yield βψ + µσ = 0 , (130)and the variations on σ yield σ = − yβ ( µξ + α ) . (131)This returns the equations (111), (113) for the mean flow.Together with Eq. (117), we recover the Gibbs stategiven by Eq. (112). Considering now the second varia-tions of entropy given by Eq. (128), we find that a criticalpoint of (124) is a maximum of S at fixed microscopic en-ergy, helicity and angular momentum iff − β (cid:18)Z δψδξ dydz + Z ( δσ ) y dydz (cid:19) − µ Z δξδσ dydz − β (cid:18)Z ψδξ dydz + Z σδσ y dydz (cid:19) ≤ , (132)for all perturbations δξ and δσ that conserve helicity andangular momentum at first order (the conservation of mi-croscopic energy has been automatically taken into ac-count in our formulation). The stability criterion (132)is equivalent to Eq. (115) but it is much simpler becauseit depends only on the perturbations δξ and δσ insteadof the perturbations δρ of the full distribution of angularmomentum. In fact, the stability condition (132) can befurther simplified. Indeed, using Eqs. (130) and (131),we find that the last term in parenthesis can be written Z ψδξ dydz + Z σδσ y dydz = − µβ Z (cid:0) σδξ + ξδσ (cid:1) dydz − αβ Z δσ dydz, (133)and it vanishes since the helicity and the angular mo-mentum are conserved at first order so that δH = R (cid:0) σδξ + ξδσ (cid:1) dydz = 0 and δI = R δσ dydz = 0. There-fore, a critical point of (124) is a maximum of entropy atfixed microscopic energy, helicity and angular momentumiff − β (cid:18)Z δψδξ dydz + Z ( δσ ) y dydz (cid:19) − µ Z δξδσ dydz ≤ , (134)for all perturbations δξ and δσ that conserve helicity andangular momentum at first order. In fact, this stabilitycondition can be obtained more rapidly if we remark thatthe maximization problem (124) is equivalent to the min-imization of the macroscopic energy at fixed helicity andangular momentum (see Sec. VI C).5 C. Equivalence with the minimum energy principle
Since ln( x ) is a monotonically increasing function, it isclear that the maximization problem (124) is equivalentto min σ,ξ { E c.g. [ σ, ξ ] | H, I } , (135)with E c.g. = 12 Z ξψ dydz + Z σ y dydz, (136) H = Z ξ σ dydz, (137) I = Z σ dydz. (138)We have the equivalence(135) ⇔ (124) ⇔ (105) . (139)Therefore, the maximization of entropy at fixed micro-scopic energy, helicity and angular momentum is equiv-alent to the minimization of macroscopic energy at fixedhelicity and angular momentum. The solution of (105) isgiven by Eq. (117) where σ is solution of (135). There-fore (105) and (135) are equivalent but (135) is easierto solve because it is expressed in terms of σ instead of ρ . Our approach therefore provides a justification of theminimum energy principle in terms of statistical mechan-ics. Note that, according to (23), the principle (135) alsoassures that the mean flow associated with the statisticalequilibrium state is nonlinearly dynamically stable withrespect to the axisymmetric Euler equations.The critical points of (135) are given by the variationalproblem δE c.g. + µδH + αδI = 0 . (140)The variations on ξ yield ψ + µσ = 0 , (141)and the variations on σ yield σ = − y ( µξ + α ) . (142)This returns Eqs. (111) and (113) for the mean flow (upto a trivial redefinition of µ and α ). Together with Eq.(117), we recover the Gibbs state given by Eq. (112). Onthe other hand, this state is a minimum of E c.g. at fixed H and I iff12 Z δψδξ dydz + Z ( δσ ) y dydz + µ Z δξδσ dydz ≥ , (143) for all perturbations δξ and δσ that conserve helicity andangular momentum at first order. This is equivalent tothe criterion given by Eq. (134) as it should.We have thus shown the equivalence between the max-imization of Boltzmann entropy at fixed helicity, angularmomentum and fine-grained energy with the minimiza-tion of coarse-grained energy at fixed helicity and angu-lar momentum. This equivalence has been shown herefor global maximization. In Appendix C, we prove theequivalence for local maximization by showing that thestability criteria (115) and (143) are equivalent. D. Equivalence with the canonical ensemble
The basic maximization problem (105) is associatedwith the microcanonical ensemble since the energy E f.g. is fixed. We could also introduce a canonical ensemblewhere the inverse temperature β is fixed by making aLegendre transform J = S − βE f.g. of the entropy withrespect to the energy [52]. The corresponding maximiza-tion problem ismax ρ,ξ { J [ ρ ] | H, I, Z ρdη = 1 } . (144)A solution of (144) is always a solution of the more con-strained dual problem (105) but the reciprocal is wrongin case of ensembles inequivalence. In the present case,however, we shall show that the microcanonical ensem-ble (105) and the canonical ensemble (144) are equiva-lent. This is because the fluctuations of the energy arequadratic.To solve the maximization problem (144) we can pro-ceed in two steps. We first maximize J at fixed H , I , R ρ dη = 1 and σ ( r ) = R ρη dη and ξ ( r ). This is equiv-alent to maximizing ˜ J = S − β R ρ η y dydzdη at fixed R ρ dη = 1 and σ ( r ) = R ρη dη . This leads to the optimaldistribution (117) where β is now fixed. This is clearlythe global maximum of ˜ J with the previous constraints.Using this optimal distribution, we can now express thefree energy in terms of ξ and σ by writing J [ ξ, σ ] = J [ ρ ].After straightforward calculations, we obtain J = − βE c.g. , (145)up to some constant terms (recall that β is a fixed param-eter in the present “canonical” situation). In the secondstep, we have to solve the maximization problemmax σ,ξ { J [ σ, ξ ] | H, I } . (146)Finally, the solution of (144) is given by (117) where σ is determined by (146). Therefore, the canonical varia-tional principle (144) is equivalent to (146). On the otherhand, since β >
0, the maximization problem (146) withEq. (145) is equivalent to (135). Since we have provenpreviously that (135) is equivalent to the microcanonicalvariational principle (105), we conclude that the micro-canonical and canonical ensembles are equivalent.6
VII. CONCLUSION
In the present paper, we have constructed a simpli-fied thermodynamic approach of the axisymmetric Eu-ler equations so as to describe its statistical equilibriumstates. This predicts the mean field at metaequilibriumand the fluctuations around it. We have considered twomean field theories. In the first one, we have ignored thefluctuations of vorticity. In that case, we have found thatthe fluctuations of angular momentum are Gaussian andthat the mean flow is in a Beltrami state. Furthermore,we have proven that the maximization of entropy at fixedhelicity, angular momentum and microscopic energy isequivalent to the minimization of macroscopic energy atfixed helicity and angular momentum. This provides ajustification of this selective decay principle from statis-tical mechanics. These results are very similar to thecase of 2D turbulence if we make the analogy betweenthe angular momentum (axisymmetric) and the vorticity(2D). Indeed, in the simplified statistical approach of the2D Euler equations developed in [22], the fluctuations ofvorticity are Gaussian and the mean flow is characterizedby a linear ω − ψ relationship. Furthermore, it has beenproven that the maximization of entropy at fixed energy,circulation and microscopic enstrophy is equivalent to theminimization of macroscopic enstrophy at fixed energyand circulation. This provides a justification of the min-imum enstrophy principle from statistical mechanics. Inthe second mean field theory, we have ignored the fluc-tuations of angular momentum. In that case, we havefound again that the fluctuations of potential vorticityare Gaussian and that the mean flow is in a Beltramistate. We have also observed an interesting signature ofvorticity stretching, via divergency of the variance of thevorticity fluctuations with increasing number of degreesof freedom. Overall, the variance of fluctuations providesa measure of the number of degrees of freedom and ofthe statistical temperature(s) of turbulence, and allowsto check Fluctuation-Dissipation Relations (FDR).The question is whether these results are applicable toa laboratory flow such as von K´arm´an flow. On the onehand, several basic hypotheses are not satisfied in the VKflow: it is a dissipative, forced flow, and instantaneous ve-locity fields are not axisymmetric. On the other hand,one observes that in a stationary state, dissipation andforcing balance globally, and may be neglected locally,at least within an inertial range of scales, and the meanflow is axisymmetric. This motivated experimental testsof the equilibrium and fluctuations relations, reported in[9] in a large Reynolds number von K´arm´an flow. It hasbeen found that the observed stationary states are welldescribed by the equilibrium states of the Euler equa-tions, and that both the velocity and vorticity fluctu-ations are approximately uniform over the box. Thesefluctuations depend on three unknowns: the mean fieldstatistical temperatures 1 /β ξ and 1 /β σ , and the effectivenumber of degrees of freedom N tot . One can thereforeuse the experimental measurements to estimate these pa- rameters under a supplementary assumption. Assuming N tot = 1, one finds that β σ and β ξ differ by an order ofmagnitude [9], an intriguing result that may come fromthe out-of-equilibrium character of the turbulence. Onthe other hand, assuming β σ = β ξ one finds that N tot isof the order of 4 to 6, a rather small number for a turbu-lent flow. This may be due to strong correlations withinthe flow, that effectively reduces the number of degreesof freedom.Altogether, these results are an indication that ther-modynamics of Euler axisymmetric flows can bring newinteresting information about real flows. Indeed, our sta-tistical theory of axisymmetric flows that can account forcertain experimental results reported in [7, 9]. In a forth-coming communication [21] we shall explain from thisapproach a turbulent bifurcation that has been observedin a von K´arm´an flow [45]. Despite its good agreementwith experiments, several criticisms can be made to ourapproach. For example, in contrast with what is hypothe-sized or derived in the paper, it is indeed usually believedthat the statistics of fluctuations in turbulence are essen-tially non-Gaussian, that the mean-field approximationis not satisfactory, and that the forcing and dissipationare important in the process. We remark that these prop-erties have been observed for homogeneous and isotropic(three dimensional) turbulence. In our 2 .
5D situation,these general results may not be correct anymore, sincethe dominant dynamical processes in both systems areprobably different, in particular because of the quasi-2Dnature of axisymmetric turbulence. In our case, it is notunlikely that the distribution of angular momentum isGaussian (or close to Gaussian), like the distribution ofthe velocity components in 3D turbulence. However, thedistribution of azimuthal vorticity, which is a derivative,may not be Gaussian because of intermittency. Unfortu-nately, we are not able to check these predictions exper-imentally due to a lack of statistics.We have applied the maximum entropy principle tothe Euler equation. Alternatively, Adzhemyan & Nal-imov [46, 47] have applied this principle to the stochasticNavier-Stokes equation. The renormalization group ap-proach was used instead of the mean-field approximation,a non-Gaussian distribution was obtained and the Kol-mogorov spectrum was derived for the inertial range. Itwould be interesting to extend their approach to our 2 . Appendix A: Generalization and link with otherresults
In the present paper, we have considered a restrictedclass of flows for which the only invariants are E , H = H and I = I . The complete generalization to arbitrary in-variants of axisymmetric Euler equations ( E , H F , I G )7remains an unsolved problem. There are however somespecial cases where we can perform the maximizationproblem and find equilibrium distributions. We list thesecases below, and make connection with previous results.
1. Conservation of E c.g. , H = H , Γ = H , I = I and I f.g.n> Leprovost et al. [6] have considered the maximizationproblemmax ρ,ξ { S [ ρ ] | E c.g. , H, Γ , I, I f.g.n> , Z ρdη = 1 } , (A1)where S [ ρ ] is the mixing entropy (40) and I f.g.n> = R ρη n dydzdη . If we make a Legendre transform of theentropy with respect to the fragile constraints, we obtainthe reduced maximization problemmax ρ,ξ { S χ [ ρ ] | E c.g. , H, Γ , I, Z ρdη = 1 } , (A2)with S χ [ ρ ] = S [ ρ ] − X n> α n I f.g.n . (A3)Explicitly S χ [ ρ ] = − Z ρ ln (cid:20) ρχ ( η ) (cid:21) dydzdη, (A4)where χ ( η ) = e − P n> α n η n . Proceeding as in [42], we canshow that the maximization problem (A2) is equivalentto max σ,ξ { S [ σ ] | E c.g. , H, Γ , I } , (A5)where S [ σ ] is the generalized entropy S = − Z C ( σ ) d r , C ( σ ) = − Z σ [(ln ˆ χ ) ′ ] − ( − x ) dx. (A6)The critical points of (A5) are given by βψ = − µσ − γ, (A7) − C ′ ( σ ) = β σ y + µξ + α. (A8)The solutions of (A2) and (A5) are always solutions of(A1) but the reciprocal is wrong in case of ensemble in-equivalence. Thus (A2) ⇔ (A5) ⇒ (A1). If we con-sider the particular case where α n = 0 for n = 2 and α = 0, then, proceeding as in [42], we find that the fluc-tuations of angular momentum are Gaussian with vari-ance σ = 1 / (2 α ) and that the generalized entropy is S = − σ Z σ d r = − σ I c.g. . (A9) Since σ >
0, the generalized entropy is proportional tominus I c.g. . The corresponding critical points βψ = − µσ − γ, (A10) − σσ = β σ y + µξ + α, (A11)are steady states of the axisymmetric Euler equationscorresponding to f ( ψ ) linear and g ( ψ ) linear but not con-stant. Therefore, the statistical approach of Leprovost etal. [6] based on (A1) does not lead to Beltrami states(corresponding to f linear and g constant) contrary tothe statistical approach developed in the present paper.Finally, if we consider the maximization problemmax ρ,ξ { S [ ρ ] | E c.g. , H, Γ , I, I f.g. , Z ρdη = 1 } , (A12)where only the quadratic integral I f.g. is conservedamong the set of fragile constraints, and proceed as in[22], we find that (A1) is equivalent tomin σ,ξ { I c.g. [ σ ] | E c.g. , H, Γ , I } . (A13)Since (A13) is equivalent to (A5) with (A9), hence to(A2), we also conclude that, in the specific case whereonly I f.g. is conserved among the set of fragile con-straints, (A12) is equivalent to (A2) with α n = 0 for n = 2.
2. Conservation of E c.g. , H = H , Γ = H and I = I We consider the maximization problemmax ρ,ξ { S [ ρ ] | E c.g. , H, Γ , I, Z ρdη = 1 } , (A14)The variations over ξ give βψ = − µσ − γ, (A15)and the variations over ρ give the exponential distribu-tion ρ = 1 Z e − ( β σ y + µξ + α ) η . (A16)Since this distribution is not normalizable, we must im-pose some bounds on the angular momentum and weshall assume − λ < σ < λ (symmetric). In that case, wehave σ = λL (cid:20) − λ (cid:18) β σ y + µξ + α (cid:19)(cid:21) , (A17)where L ( x ) = tan − ( x ) − x , (A18)8is the Langevin function [48]. We see that ( ξ, σ ) is asteady state of the axisymmetric Euler equations. If weconsider the maximization problemmax ρ,σ { S [ ρ ] | E c.g. , H, Γ , I, Z ρdν = 1 } , (A19)we find symmetric results but, in that case, ( ξ, σ ) is not a steady state of the axisymmetric Euler equations.
3. Conservation of E f.g. , H = H and I = I In the present paper, we have considered the maxi-mization problemmax ρ,ξ { S [ ρ ] | E f.g. , H, I, Z ρdη = 1 } . (A20)The variations over ξ give βψ + µσ = 0 (A21)and the variations over ρ give the Gaussian distribution ρ = 1 Z e − βη y − ( µξ + α ) η . (A22)We have σ = − yβ (cid:0) µξ + α (cid:1) , σ = 2 yβ . (A23)In that case ( ξ, σ ) is a steady state of the Euler equa-tions with f linear and g constant (Beltrami state). Thestream function can be expressed in terms of Bessel func-tions. Furthermore, we have shown that (A20) is equiv-alent to min σ,ξ { E c.g. [ σ ] | H, I } . (A24)
4. General case: conservation of E f.g. , H = H , I , H f.g.n> , I f.g.n> Let us consider the problemmax ρ,ξ { S [ ρ ] | E f.g. , H, I, H f.g.n> , I f.g.n> , Z ρdη = 1 } , (A25)generalizing the one studied in the present paper. Thevariations over ξ give βψ + µσ + X n> µ n σ n = 0 , (A26)and the variations over ρ give ρ = 1 Z e − P n> α n η n e − P n> µ n ξη n e − βη y e − ( µξ + α ) η . (A27)However, it is difficult to be more explicit. Therefore, weshall consider simpler problems.
5. Conservation of E f.g. , H = H , I = I , and I f.g. We consider the maximization problemmax ρ,ξ { S [ ρ ] | E f.g. , H, I, I f.g. , Z ρdη = 1 } . (A28)The variations over ξ give βψ + µσ = 0 , (A29)and the variations over ρ give the Gaussian distribution ρ = 1 √ πσ e − ( η − σ )22 σ , (A30)with σ = − µξ + α α + β y ) , σ = 12( α + β y ) . (A31)In that case ( ξ, σ ) is a steady state of the Euler equa-tions with f and g linear. The stream function can beexpressed in terms of Whittaker functions [6]. If we makea Legendre transform of the entropy with respect to thefragile constraints, we obtain the reduced maximizationproblem max ρ,ξ {S [ ρ ] | H, I, Z ρdη = 1 } , (A32)with S = S − βE f.g. − α I f.g. . (A33)We proceed as in [42]. We first maximize S at fixed H , I ,normalization and σ = R ρη dη . This yields an optimaldensity ρ ( r , η ) given by Eq. (A30) where σ is givenby Eq. (A31)-b. Then, we find that the maximizationproblem (A32) is equivalent tomax σ,ξ { S [ σ, ξ ] | H, I } , (A34)with the generalized entropy S [ σ, ξ ] ≡ S [ ρ ]. An explicitcalculation leads to S [ σ, ξ ] = − βE c.g. − α I c.g. . (A35)This is a sort of “mixed” case between the one studied inthe main part of the paper (leading to the minimizationof E c.g. ) and the one discussed at the beginning of thisAppendix (leading to the minimization of I c.g. ). Thecritical points of (A34) return Eqs. (A29) and (A31)-a. Furthermore, a solution of (A32) or (A34) is alwaysa solution of (A28) but the reciprocal is wrong in caseof ensemble inequivalence. We have (A32) ⇔ (A34) ⇒ (A28).9
6. Conservation of E f.g. , H = H , I = I , H f.g. and I f.g. We consider the maximization problemmax ρ,ξ { S [ ρ ] | E f.g. , H, I, H f.g. , I f.g. , Z ρdη = 1 } , (A36)where S [ ρ ] is the mixing entropy (40) and H f.g.n> = R ξρη n dydzdη . The variations over ξ give βψ + µσ + µ σ = 0 (A37)and the variations over ρ give the Gaussian distribution ρ = 1 Z e − α η e − µ ξη e − βη y e − ( µξ + α ) η . (A38)We have σ = − µξ + α α + µ ξ + β y ) , (A39) σ = 12( α + µ ξ + β y ) . (A40) Appendix B: Detailed proof of inequality (39)
We consider the minimization problemmin ξ,σ { E [ ξ, σ ] | H, I } , (B1)with E = 12 Z ξψ dydz + 14 Z σ y dydz, (B2) H = Z ξσ dydz, (B3) I = Z σ dydz. (B4)We shall look for (local) minima of energy at fixed helicityand angular momentum. We proceed as in [25, 50]. Thevariations of these functionals up to second order are∆ E = Z ψδξ dydz + 12 Z δξδψ dydz + 12 Z σδσy dydz + 14 Z ( δσ ) y dydz, (B5)∆ H = Z ξδσ dydz + Z σδξ dydz + Z δξδσ dydz, (B6) ∆ I = Z δσ dydz. (B7)The critical points satisfy the variational principle for thefirst variations δE + µδH + αδI = 0 . (B8)Taking the variations over ξ and σ , we obtain ψ + µσ = 0 , (B9) σ y + µξ + α = 0 . (B10)A minimum of energy corresponds to ∆ E >
0. InsertingEqs. (B9) and (B10) in Eq. (B5), we find that∆ E = − µ Z σδξ dydz + 12 Z δξδψ dydz − µ Z ξδσ dydz − α Z δσ dydz + 14 Z ( δσ ) y dydz. (B11)Then, using Eqs. (B6) and (B7) with ∆ H = ∆ I = 0, weobtain∆ E = 12 Z δξδψ dydz + Z ( δσ ) y dydz + µ Z δξδσ dydz. (B12)Therefore, a critical point of (B1) is a (local) minimumof energy at fixed helicity and angular momentum iff12 Z δξδψ dydz + Z ( δσ ) y dydz + µ Z δξδσ dydz ≥ , (B13)for all perturbations δσ and δξ that conserve helicity andangular momentum at first order. This amounts to hav-ing δ ( E + µH + αI ) > H and I at first order. Appendix C: Equivalence between (115) and (143)
In Sec. VI, we have shown the equivalence of (105)and (135) for global maximization. In this Appendix, weshow the equivalence of (105) and (135) for local max-imization, i.e. ρ ( r , η ) is a (local) maximum of S [ ρ ] atfixed E f.g. , H , I and normalization if, and only if, thecorresponding coarse-grained distribution of angular mo-mentum σ ( r ) is a (local) minimum of E c.g. [ σ, ξ ] at fixed H and I . To that purpose, we show the equivalence be-tween the stability criteria (115) and (143). We use ageneral method similar to the one used in [22, 42, 49, 50]in related problems.We shall determine the optimal perturbation δρ ∗ ( r , η )that maximizes δ J [ δρ ] given by Eq. (115) with theconstraints δσ = R δρη dη , δE f.g. = R ψδξ dydz +0 R δρ η y dηdydz = 0 and R δρ dη = 0, where δσ ( r ) and δξ ( r ) are prescribed (they are only ascribed to conserve H and I at first order). Since the specification of δσ and δξ (hence δψ ) determine the second and third integralsin Eq. (115), we can write the variational problem in theform δ (cid:18) − Z ( δρ ) ρ dydzdη (cid:19) − Z λ ( r ) δ (cid:18)Z δρη dη (cid:19) dydz − ˜ µδ (cid:18)Z δρ η y dydzdη (cid:19) − Z ζ ( r ) δ (cid:18)Z δρ dη (cid:19) dydz = 0 , (C1)where λ ( r ), ˜ µ and ζ ( r ) are Lagrange multipliers. Thisgives δρ ∗ ( r , η ) = − ρ ( r , η ) (cid:20) ˜ µ η y + λ ( r ) η + ζ ( r ) (cid:21) , (C2)and it is a global maximum of δ J [ δρ ] with the previ-ous constraints since δ ( δ J ) = − R ( δ ( δρ )) ρ dydzdη < δρ so their second varia-tions vanish). The Lagrange multipliers are determinedfrom the above-mentioned constraints. The constraints R δρ dη = 0 and δσ = R δρη dη lead to ζ ( r ) + λ ( r ) σ ( r ) + ˜ µ σ ( r )4 y = 0 , (C3) ζ ( r ) σ ( r ) + λ ( r ) σ ( r ) + ˜ µ σ ( r )4 y = − δσ ( r ) . (C4)Now, the state ρ ( r , η ) corresponds to the gaussian dis-tribution (112). Therefore, we have the well-known re-lations σ ( r ) = σ ( r ) + σ and σ ( r ) = σ ( r ) + 3 σ ( r ) σ where σ = 2 y/β . Substituting these relations in Eqs.(C3) and (C4), and solving for λ ( r ) and ζ ( r ), we obtain λ ( r ) = − β y δσ ( r ) − ˜ µ y σ ( r ) , (C5) ζ ( r ) = β y σ ( r ) δσ ( r ) + ˜ µ y σ ( r ) − ˜ µ β . (C6)Therefore, the optimal perturbation (C2) can be rewrit-ten δρ ∗ = − ρ (cid:20) − β y δσ ( η − σ ) + ˜ µ (cid:26) y ( η − σ ) − β (cid:27)(cid:21) . (C7)The Lagrange multiplier ˜ µ is determined by substi-tuting this expression in the constraint R ψδξ dydz + R δρ η y dydzdη = 0. Using the well-known identity σ ( r ) = σ ( r ) + 6 σ σ ( r ) + 3 σ valid for a gaussian dis-tribution, we obtain after some simplifications˜ µ = 2 β (cid:18)Z ψδξ dydz + Z σ y δσ dydz (cid:19) . (C8) Therefore, the optimal perturbation (C2) is given by Eq.(C7) with Eq. (C8). Since this perturbation maximizes δ J [ δρ ] with the above-mentioned constraints, we have δ J [ δρ ] ≤ δ J [ δρ ∗ ]. Explicating δ J [ δρ ∗ ] using Eqs. (C7)and (C8), we obtain after simple calculations δ J [ δρ ] ≤ − β (cid:18)Z δψδξ dydz + Z ( δσ ) y dydz (cid:19) − µ Z δξδσ dydz − β (cid:18)Z ψδξ dydz + Z σδσ y dydz (cid:19) . (C9)The r.h.s. returns the functional appearing in Eq. (132).We have already explained in Sec. VI B that for the classof perturbations that we consider ( δH = δI = 0) the lastterm in parenthesis vanishes. Therefore, the foregoinginequality can be rewritten δ J [ δρ ] ≤ − β (cid:18)Z δψδξ dydz + Z ( δσ ) y dydz (cid:19) − µ Z δξδσ dydz, (C10)where the r.h.s. is precisely the functional appearing inEq. (143). Furthermore, there is equality in Eq. (C10)iff δρ = δρ ∗ . This proves that the stability criteria (115)and (143) are equivalent. Indeed: (i) if inequality (143)is fulfilled for all perturbations δσ and δξ that conservehelicity and angular momentum at first order, then ac-cording to Eq. (C10), we know that inequality (115) isfulfilled for all perturbations δρ and δξ that conserve he-licity, angular momentum, fine-grained energy and nor-malization at first order; (ii) if there exists a perturbation δσ ∗ that violates inequality (143), then the perturbation δρ ∗ given by Eq. (C7) with Eq. (C8) and δσ = δσ ∗ vio-lates (115). In conclusion, the stability criteria (115) and(143) are equivalent. Appendix D: Relaxation equations1. Relaxation equations associated with themaximization problem (105)
Like in classical statistical physics, it may be interest-ing to derive relaxation equations towards the equilib-rium states so as to be able to describe dynamical, non-stationary, regimes. On a practical point of view, theserelaxation equations can also provide a useful numericalalgorithm to solve the maximization problem (105) andbe sure that we select entropy maxima (not minima orsaddle points). We follow the methodology described in[25]. We introduce a current of probability in the space ofangular momentum fluctuations η and construct a set ofrelaxation equations that increase S [ ρ ] while conserving E f.g. , I and H using a Maximum Entropy ProductionPrinciple (this can be viewed as the variational formula-1tion of Onsager’s linear thermodynamics). The dynami-cal equations that we consider can be written as ∂ξ∂t + u · ∇ ξ = ∂∂z σ y ! + X, (D1) ∂ρ∂t + u · ∇ ρ = − ∂J∂η , (D2)where X and J are two unknown quantities to be chosenso as to increase S [ ρ ] while conserving E f.g. , H and I . Inthe second equation, the local normalization R ρdη = 1 issatisfied provided that J → η → ±∞ . MultiplyingEq. (D2) by η and integrating on all the levels, we get ∂σ∂t + u · ∇ σ = Z Jdη ≡ Y. (D3)Next, multiplying Eq. (D2) by η and integrating on allthe levels, we obtain ∂σ ∂t + u · ∇ σ = 2 Z Jηdη. (D4)From Eqs. (D3) and (D4), we find that ∂σ ∂t + u · ∇ σ = 2 Z J ( η − σ ) dη. (D5)The time variations of S [ ρ ] are given by˙ S = − Z Jρ ∂ρ∂η dydzdη, (D6)while those of the invariants are given by˙ E f.g. = 0 = Z Xψdydz + Z J η y dydzdη, (D7)˙ H = 0 = Z Xσdydz + Z Jξdydzdη, (D8)˙ I = 0 = Z Jdydzdη. (D9)Following the Maximum Entropy Production Principle,we maximize ˙ S with ˙ E f.g. = ˙ H = ˙ I = 0 and the addi-tional constraints X ≤ C ξ ( r , t ) , Z J ρ dη ≤ C ( r , t ) . (D10)The variational principle can be written in the form δ ˙ S − β ( t ) δ ˙ E f.g. − µ ( t ) δ ˙ H − α ( t ) δ ˙ I − Z D ( r , t ) δ (cid:18)Z J ρ dη (cid:19) dydz − Z χ ( r , t ) δ (cid:18) X (cid:19) dydz = 0 , (D11)where β ( t ), µ ( t ), α ( t ), D ( r , t ) and χ ( r , t ) are time de-pendent Lagrange multipliers associated with the con-straints. This leads to the following optimal quantities J = − D (cid:20) ∂ρ∂η + ρ (cid:18) β ( t ) η y + µ ( t ) ξ + α ( t ) (cid:19)(cid:21) , (D12) X = − χ ( β ( t ) ψ + µ ( t ) σ ) . (D13) Therefore, the relaxation equation for the distribution ofangular momentum is ∂ρ∂t + u · ∇ ρ = ∂∂η (cid:26) D (cid:20) ∂ρ∂η + ρ (cid:18) β ( t ) η y + µ ( t ) ξ + α ( t ) (cid:19)(cid:21)(cid:27) . (D14)Integrating Eq. (D12) on η we get Y = − D (cid:18) β ( t ) σ y + µ ( t ) ξ + α ( t ) (cid:19) . (D15)Inserting expressions (D13) and (D15) into Eqs. (D1) and(D3) leads to the following relaxation equations for themean flow ∂ξ∂t + u · ∇ ξ = ∂∂z σ y ! − χ ( β ( t ) ψ + µ ( t ) σ ) , (D16) ∂σ∂t + u · ∇ σ = − D (cid:18) β ( t ) σ y + µ ( t ) ξ + α ( t ) (cid:19) . (D17)A relaxation equation can also be written for the centeredvariance σ . Using Eqs. (D5) and (D12), we obtain ∂σ ∂t + u · ∇ σ = 2 D (cid:18) − β ( t ) σ y (cid:19) . (D18)Equations (D17) and (D18) can be used to evaluate theevolution of σ = σ + σ . The Lagrange multipliersevolve in time so as to satisfy the constraints. Substitut-ing Eqs. (D12) and (D13) in Eqs. (D7), (D8) and (D9),we obtain the algebraic equations (cid:10) χψ (cid:11) + * D σ y +! β ( t ) + (cid:18) h χσψ i + (cid:28) D ξσ y (cid:29)(cid:19) µ ( t )+ (cid:28) D σ y (cid:29) α ( t ) = (cid:28) D y (cid:29) , (D19) (cid:18) h χψσ i + (cid:28) D σξ y (cid:29)(cid:19) β ( t ) + (cid:16)(cid:10) χσ (cid:11) + D Dξ E(cid:17) µ ( t )+ (cid:10) Dξ (cid:11) α ( t ) = 0 , (D20) (cid:28) D σ y (cid:29) β ( t ) + (cid:10) Dξ (cid:11) µ ( t ) + α ( t ) h D i = 0 . (D21)The coefficients D and χ , which can depend on y and z ,are not determined by the MEPP. They can be chosenso as to forbid divergency of the first term in the r.h.s.of equation (D19).Substituting ∂ρ/∂η taken from Eq. (D12) in Eq. (D6)and using the constraints (D7)-(D9), we easily obtain˙ S = Z J Dρ dydzdη + Z X χ dydz, (D22)2so that ˙ S ≥ D and χ are both posi-tive. On the other hand ˙ S = 0 iff J = X = 0 leading tothe conditions of equilibrium (111) and (112). From Lya-punov’s direct method, we conclude that these relaxationequations tend to a maximum of entropy at fixed mi-croscopic energy, helicity and angular momentum. Notethat during the relaxation process, the distribution of an-gular momentum is not Gaussian but changes with timeaccording to Eq. (D14). The distribution is Gaussianonly at equilibrium. Therefore, these relaxation equa-tions describe not only the evolution of the mean flowbut also the evolution of the distribution of fluctuations.We stress, however, that these equations are purely phe-nomenological and that there is no compelling reasonwhy they should give an accurate description of the realdynamics. However, they can be used as a numerical al-gorithm to compute the equilibrium state correspondingto (105). Indeed, these equations can only relax towardsan entropy maximum at fixed microscopic energy, helic-ity and angular momentum, not towards a minimum ora saddle point that are linearly unstable with respect tothese equations. Remark:
In fact, we will find in [21] that there is noentropy maximum, just saddle points. In that case, thedynamical equations lead to a “collapse” at smaller andsmaller scales, similar to the Richardson energy cascadein 3D turbulence. However, we will also observe thatthe system can remain blocked in a large-scale coherentstructure (like in 2D turbulence). In the present 2.5Dsituation, this is an unstable state (saddle point of en-tropy), but it can persist for a long time if the dynamicsdoes not spontaneously develop the “dangerous” pertur-bations that destabilize it. This is because a saddle pointis unstable only for some perturbations but not for anyperturbation.
2. Relaxation equations associated with themaximization problem (124)
We shall now introduce a set of relaxation equationsassociated with the maximization problem (124). Wewrite the dynamical equations as [53]: ∂ξ∂t + u · ∇ ξ = ∂∂z (cid:18) σ y (cid:19) + X, (D23) ∂σ∂t + u · ∇ σ = Y, (D24)where X and Y are two unknown quantities, to be chosenso as to increase S [ ξ, σ ] while conserving E f.g. , H and I given by Eqs. (125), (126) and (127). The time variationsof S are˙ S = − β ( t ) (cid:18) Z ψX dydz + Z σy Y dydz (cid:19) , (D25) where β ( t ) is determined by the constraint on the micro-scopic energy leading to1 β ( t ) = 2 E f.g. − Z ξψ dydz − Z σ y dydz. (D26)On the other hand, the time variations of H and I are˙ H = 0 = Z Xσ dydz + Z Y ξ dydz, (D27)˙ I = 0 = Z Y dydz. (D28)Following the Maximum Entropy Production Princi-ple, we maximize ˙ S with ˙ I = ˙ H = 0 (the conservationof the microscopic energy has been taken into account inEq. (D26)) and the additional constraints X ≤ C ξ ( r , t ) , Y ≤ C σ ( r , t ) . (D29)The variational principle can be written in the form δ ˙ S − µ ( t ) δ ˙ H − α ( t ) δ ˙ I − Z χ ( r , t ) δ (cid:18) X (cid:19) dydz − Z D ( r , t ) δ (cid:18) Y (cid:19) dydz = 0 , (D30)and it leads to the following quantities X = − χ ( β ( t ) ψ + µ ( t ) σ ) , (D31) Y = − D (cid:18) β ( t ) σ y + µ ( t ) ξ + α ( t ) (cid:19) . (D32)Inserting expressions (D31) and (D32) into Eqs. (D23)and (D24), we obtain the relaxation equations ∂ξ∂t + u · ∇ ξ = ∂∂z (cid:18) σ y (cid:19) − χ ( β ( t ) ψ + µ ( t ) σ ) , (D33) ∂σ∂t + u · ∇ σ = − D (cid:18) β ( t ) σ y + µ ( t ) ξ + α ( t ) (cid:19) . (D34)The Lagrange multipliers evolve so as to satisfy the con-straints. Substituting Eqs. (D31) and (D32) in Eqs.(D27) and (D28), and recalling Eq. (D26), we obtain thealgebraic equations1 β ( t ) = 2 E f.g. − h ξψ i − (cid:28) σ y (cid:29) , (D35) (cid:18) h χψσ i + (cid:28) D σξ y (cid:29)(cid:19) β ( t ) + (cid:16)(cid:10) χσ (cid:11) + D Dξ E(cid:17) µ ( t )+ (cid:10) Dξ (cid:11) α ( t ) = 0 , (D36) (cid:28) D σ y (cid:29) β ( t ) + (cid:10) Dξ (cid:11) µ ( t ) + α ( t ) h D i = 0 . (D37)3Substituting ψ and σ/y taken from Eqs. (D31) and (D32)in Eq. (D25) and using the constraints (D27) and (D28),we easily obtain˙ S = Z X χ dydz + Z Y D dydz, (D38)so that ˙ S ≥ D and χ are both positive.On the other hand ˙ S = 0 iff X = Y = 0 leading to theconditions of equilibrium given by Eqs. (130) and (131).From Lyapunov’s direct method, we conclude that theserelaxation equations tend to a maximum of entropy atfixed microscopic energy, helicity and angular momen-tum.The relaxation equations (D33) and (D34) are similarto Eqs. (D16) and (D17) but the constraints determiningthe evolution of the Lagrange multipliers are different.More precisely, Eqs. (D36) and (D37) coincide with Eqs.(D20) and (D21) but Eq. (D19) has been replaced byEq. (D35). Indeed, in the present approach, the distri-bution of angular momentum is always Gaussian duringthe dynamical evolution. It is given by Eq. (117) at anytime, i.e. ρ ( r , η, t ) = (cid:18) β ( t )4 πy (cid:19) / e − β ( t )4 y ( η − σ ( r ,t )) . (D39)By contrast, in Sec. VI A, the distribution of angularmomentum changes with time. Therefore, the dynamicalevolution is different. However, in the two approaches,the equilibrium state is the same, i.e. it solves the maxi-mization problem (105). This is sufficient if we use theserelaxation equations as numerical algorithms to computethe maximum entropy state. Remark:
Using Eqs. (D23), (D24) and (D25), it iseasy to show that ˙ S = − β ( t ) ˙ E c.g. so that ˙ E c.g. ≤ β ( t ) ≥
0. Therefore, the macroscopic energy monotoni-cally decreases through the relaxation equations. This isto be expected since the maximization problem (124) isequivalent to the minimization of the macroscopic energyat fixed helicity and angular momentum (see Sec. VI C).
3. Relaxation equations associated with theminimization problem (135)
We shall introduce a set of relaxation equations asso-ciated with the minimization problem (135). We writethe dynamical equations as ∂ξ∂t + u · ∇ ξ = ∂∂z (cid:18) σ y (cid:19) + X, (D40) ∂σ∂t + u · ∇ σ = Y, (D41)where X and Y are two unknown quantities to be chosenso as to decrease E c.g. while conserving H and I . Thetime variations of E c.g. are given by˙ E c.g. = Z ψX dydz + Z σ y Y dydz. (D42) On the other hand, the time variations of H and I are˙ H = 0 = Z Xσ dydz + Z Y ξ dydz, (D43)˙ I = 0 = Z Y dydz. (D44)Following the Maximum Entropy Production Princi-ple, we maximize the dissipation ˙ E c.g. with ˙ I = ˙ H = 0and the additional constraints X ≤ C ξ ( r , t ) , Y ≤ C σ ( r , t ) . (D45)The variational principle can be written in the form δ ˙ E c.g. + µ ( t ) δ ˙ H + α ( t ) δ ˙ I + Z χ ( r , t ) δ (cid:18) X (cid:19) dydz + Z D ( r , t ) δ (cid:18) Y (cid:19) dydz = 0 , (D46)and we obtain the following quantities X = − χ ( ψ + µ ( t ) σ ) , (D47) Y = − D (cid:18) σ y + µ ( t ) ξ + α ( t ) (cid:19) . (D48)Substituting Eqs. (D47) and (D48) into Eq. (D40) and(D41) leads to the following relaxation equations ∂ξ∂t + u · ∇ ξ = ∂∂z (cid:18) σ y (cid:19) − χ ( ψ + µ ( t ) σ ) , (D49) ∂σ∂t + u · ∇ σ = − D (cid:18) σ y + µ ( t ) ξ + α ( t ) (cid:19) . (D50)The Lagrange multipliers µ ( t ) and α ( t ) evolve so as tosatisfy the constraints (D43) and (D44). SubstitutingEqs. (D47) and (D48) in Eqs. (D43) and (D44), weobtain the algebraic equations (cid:16)(cid:10) χσ (cid:11) + D Dξ E(cid:17) µ ( t ) + (cid:10) Dξ (cid:11) α ( t )+ h χψσ i + (cid:28) D σξ y (cid:29) = 0 , (D51) (cid:10) Dξ (cid:11) µ ( t ) + α ( t ) h D i + (cid:28) D σ y (cid:29) = 0 . (D52)Substituting ψ and σ/y taken from Eqs. (D47) and (D48)in Eq. (D42) and using the constraints (D43) and (D44),we easily obtain˙ E c.g. = − Z X χ dydz − Z Y D dydz, (D53)so that ˙ E c.g. ≤ D and χ are both posi-tive. On the other hand, ˙ E c.g. = 0 iff X = Y = 0 leading4to the conditions of equilibrium given by Eqs. (141) and(142). By Lyapunov’s direct method, we conclude thatthese relaxation equations tend to a minimum of macro-scopic energy E c.g. at fixed helicity and angular momen-tum. Therefore, the relaxation equations (D49,D50) canbe used as a numerical algorithm to solve the minimiza-tion problem (135). Remark: since these relaxation equations solve Eq.(32), they can also be used as a numerical algorithm toconstruct nonlinearly dynamically stable stationary solu-tions of the axisymmetric Euler equations correspondingto Beltrami states (see Secs. III and IV) independentlyof the statistical mechanics interpretation.
Appendix E: Another type of relaxation equations
In the main part of the paper, we have not taken intoaccount the conservation of circulation Γ = R ξ dydz be-cause there is no critical point of energy at fixed helicity,angular momentum and circulation (see [21]). Neverthe-less, at the level of the relaxation equations, it is possibleto take this constraint into account. We shall introducea set of relaxation equations that minimize the energy E c.g. at fixed helicity H , angular momentum I and cir-culation Γ. Since there is no energy minimum (not evena critical point of energy), these equations should havea non-trivial behavior. To derive these equations, onepossibility is to write them in the form (D40)-(D41) andintroduce Lagrange multipliers for each constraint. An-other possibility is to write them in the form ∂ξ∂t + u · ∇ ξ = ∂∂z (cid:18) σ y (cid:19) − ∇ · J ξ , (E1) ∂σ∂t + u · ∇ σ = −∇ · J σ , (E2)where J ξ and J σ are two unknown currents to be chosenso as to decrease E c.g. while conserving H . The form(E1)-(E2) guarantees the conservation of circulation andangular momentum. The time variations of E c.g. aregiven by˙ E c.g. = Z J ξ · ∇ ψ dydz + Z J σ · ∇ (cid:18) σ y (cid:19) dydz. (E3)On the other hand, the time variations of H are˙ H = 0 = Z J ξ · ∇ σ dydz + Z J σ · ∇ ξ dydz. (E4) Following the Maximum Entropy Production Princi-ple, we maximize the dissipation ˙ E c.g. with ˙ H = 0 andthe additional constraints J ξ ≤ C ξ ( r , t ) , J σ ≤ C σ ( r , t ) . (E5)The variational principle can be written in the form δ ˙ E c.g. + µ ( t ) δ ˙ H + Z D ξ ( r , t ) δ J ξ ! dydz + Z D σ ( r , t ) δ (cid:18) J σ (cid:19) dydz = 0 , (E6)and we obtain the optimal currents J ξ = − D ξ ( ∇ ψ + µ ( t ) ∇ σ ) , (E7) J σ = − D σ (cid:20) ∇ (cid:18) σ y (cid:19) + µ ( t ) ∇ ξ (cid:21) . (E8)Substituting Eqs. (E7) and (E8) into Eq. (E1) and (E2)leads to the following relaxation equations ∂ξ∂t + u · ∇ ξ = ∂∂z (cid:18) σ y (cid:19) + ∇ · [ D ξ ( ∇ ψ + µ ( t ) ∇ σ )] , (E9) ∂σ∂t + u · ∇ σ = ∇ · (cid:26) D σ (cid:20) ∇ (cid:18) σ y (cid:19) + µ ( t ) ∇ ξ (cid:21)(cid:27) . (E10)The Lagrange multiplier µ ( t ) evolves so as to satisfy theconstraint (E4). Substituting Eqs. (E7) and (E8) in Eq.(E4), we obtain µ ( t ) = − R D ξ ∇ ψ · ∇ σ dydz + R D σ ∇ (cid:16) σ y (cid:17) · ∇ ξ dydz R D ξ ( ∇ σ ) dydz + R D σ ( ∇ ξ ) dydz . (E11)Substituting ∇ ψ and ∇ ( σ/y ) taken from Eqs. (E7) and(E8) in Eq. (E3) and using the constraint (E4), we easilyobtain ˙ E c.g. = − Z J ξ D ξ dydz − Z J σ D σ dydz, (E12)so that ˙ E c.g. ≤ D ξ and D σ are bothpositive. [1] L. Onsager, Nuovo Cimento, Suppl. , 279 (1949).[2] D. Montgomery, G. Joyce, Phys. Fluids , 1139 (1974).[3] T.S. Lundgren, Y.B. Pointin, J. Stat. Phys. , 323(1977).[4] J. Miller, Phys. Rev. Lett. , 2137 (1990). [5] R. Robert and J. Sommeria, J. Fluid Mech. , 291(1991).[6] N. Leprovost, B. Dubrulle, and P.H. Chavanis, Phys.Rev. E , 046308 (2006).[7] R. Monchaux, F. Ravelet, B. Dubrulle, A. Chiffaudel, and F. Daviaud, Phys. Rev. Lett. , 124502 (2006).[8] W. Matthaeus, D. Montgomery, Ann. N.Y. Acad. Sci. , 203 (1980).[9] R. Monchaux, P.-P. Cortet, P.H. Chavanis, A. Chiffaudel,F. Daviaud, P. Diribarne and B. Dubrulle, Phys. Rev.Lett. , 174502 (2008).[10] B. Dubrulle, P.H. Chavanis, A. Chiffaudel, G. Collette,P.-P. Cortet, F. Daviaud, P. Diribarne, R. Monchaux, A.Naso, S. Thalabard, in preparation[11] J. Duchon and R. Robert, Nonlinearity (1), 249 (2000).[12] Y.B. Pointin, T.S. Lundgren, Phys. Fluids , 1459(1976).[13] A.C. Ting, H.H. Chen, Y.C. Lee, Physica D , 37(1987).[14] B.N. Kuvshinov, T.J. Schep, Phys. Fluids , 3282(2000).[15] W.H. Matthaeus, W.T. Stribling, D. Martinez, S.Oughton, D. Montgomery, Physica D , 531 (1991).[16] D. Montgomery, W.H. Matthaeus, W.T. Stribling, D.Martinez, S. Oughton, Phys. Fluids A , 3 (1992).[17] S.J. Li, D. Montgomery, Phys. Lett. A , 281 (1996); , 461 (1996).[18] S.J. Li, D. Montgomery, W.B. Jones, Theor. & Comp.Fluid Dyn. , 167 (1997).[19] D.J. Rodgers, S. Servidio, W.H. Matthaeus, D.C. Mont-gomery, T.B. Mitchell, T. Aziz, Phys. Rev. Lett. ,244501 (2009).[20] G.J.F. van Heijst, H.J.H. Clercx, D. Molenaar, J. FluidMech. , 411 (2006).[21] A. Naso, S. Thalabard, G. Collette, R. Monchaux, P.H.Chavanis and B. Dubrulle, [arXiv1002.2711].[22] A. Naso, P.H. Chavanis, B. Dubrulle, [arXiv0912.5098].[23] D.D. Holm, J.E. Marsden, T. Ratiu, A. Weinstein, Phys.Rep. , 2 (1985).[24] R.S. Ellis, K. Haven, B. Turkington, Nonlin. , 239(2002)[25] P.H. Chavanis, Eur. Phys. J. B , 73 (2009).[26] R.S. Ellis, K. Haven, B. Turkington, J. Stat. Phys. ,999 (2000).[27] P.H. Chavanis, Phys. Rev. E , 036108 (2003).[28] J.B. Taylor, Phys. Rev. Lett. , 1139 (1974).[29] L. Woltjer, Proc. Nat. Acad. Sci. , 489 (1958).[30] D. Montgomery, L. Turner, G. Vahala, Phys. Fluids ,757 (1978).[31] F.P. Bretherton, D.B. Haidvogel, J. Fluid. Mech. , 129(1976)[32] C.E. Leith, Phys. Fluid. , 1388 (1984)[33] X.W. Shan, D. Montgomery, H.D. Chen, Phys. Rev. A , 6800 (1991). [34] D. Montgomery, X. Shan, in Small-Scale Structures inThree-Dimensional Hydrodynamic and Magnetohydrody-namic Turbulence,
Ed. by M. Meneguzzi and A. Pouquet(Berlin: Springer Verlag 1995) pp. 241-254.[35] T. Padmanabhan, Phys. Rep. , 285 (1990)[36] P.H. Chavanis, Int. J. Mod. Phys. B , 3113 (2006)[37] E. T. Jaynes, Phys. Rev. , 620 (1957)[38] P.H. Chavanis, J. Sommeria, J. Fluid. Mech. , 267(1996)[39] R.H. Kraichnan, J. Fluid. Mech. , 155 (1975)[40] R. Salmon, G. Holloway and M. C. Hendershott, J. FluidMech. , 691 (1976).[41] F. Ravelet, M. Berhanu, R. Monchaux, S. Aumaitre, A.Chiffaudel, F. Daviaud, B. Dubrulle, M. Bourgoin, Ph.Odier, N. Plihon, J.-F. Pinton, R. Volk, S. Fauve, N.Mordant, and F. Petrelis, Phys. Rev. Lett. , 074502(2008).[42] P.H. Chavanis, A. Naso, B. Dubrulle, [arXiv0912.5096].[43] P.H. Chavanis, Physica D , 257 (2005).[44] F. Bouchet, Physica D , 1976 (2008).[45] F. Ravelet, L. Mari´e, A. Chiffaudel, F. Daviaud, Phys.Rev. Lett. , 164501 (2004).[46] L.T. Adzhemyan, M.Y. Nalimov, Theor. Math. Phys. ,532 (1992)[47] L.T. Adzhemyan, M.Y. Nalimov, Theor. Math. Phys. ,872 (1993)[48] B. Turkington, Commun. Pure Appl. Math. , 781(1999).[49] P.H. Chavanis, [arXiv:1002.0291]; P.H. Chavanis, L.Delfini, Phys. Rev. E , 1 (2010)[50] A. Campa, P.H. Chavanis, [arXiv:1003.2378][51] In this paper, we shall not take into account the conser-vation of circulation Γ = R ξ dydz at equilibrium becausethere is no critical point of energy at fixed helicity, angu-lar momentum and circulation: if the Lagrange multiplier γ associated with the conservation of Γ is non-zero, thedifferential equation resulting from the variational princi-ple δE − µδH − αδI − γδ Γ = 0 presents some divergenciesat r = 0 (see [21]). However, in Appendix E, we presentdynamical equations that dissipate energy at fixed helic-ity, angular momentum and circulation.[52] This can also be viewed as a Legendre transform withrespect to the fragile constraint in the spirit of [24, 25,43, 44].[53] In the present situation, σ is given at each time by Eq.(118) where β ( t ) is given by Eq. (D26). Since σ /y doesnot depend on z , we have written σ instead of σ2